Modular Forms, Hecke Operators, and Modular Abelian Varieties · 1.3 Modular forms 5 1.3 Modular...

Modular Forms, Hecke Operators, and Modular

Abelian Varieties

Kenneth A. Ribet William A. Stein

December 9, 2003

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1 The Main objects 31.1 Torsion points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 The Tate module . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Galois representations . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Hecke operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Modular representations and algebraic curves 72.1 Arithmetic of modular forms . . . . . . . . . . . . . . . . . . . . . 72.2 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Parity Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Conjectures of Serre (mod ` version) . . . . . . . . . . . . . . . . . 92.5 General remarks on mod p Galois representations . . . . . . . . . . 92.6 Serre’s Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.7 Wiles’s Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Modular Forms of Level 1 133.1 The Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Some Examples and Conjectures . . . . . . . . . . . . . . . . . . . 143.3 Modular Forms as Functions on Lattices . . . . . . . . . . . . . . . 153.4 Hecke Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.5 Hecke Operators Directly on q-expansions . . . . . . . . . . . . . . 17

3.5.1 Explicit Description of Sublattices . . . . . . . . . . . . . . 183.5.2 Hecke operators on q-expansions . . . . . . . . . . . . . . . 193.5.3 The Hecke Algebra and Eigenforms . . . . . . . . . . . . . . 203.5.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.6 Two Conjectures about Hecke Operators on Level 1 Modular Forms 21

iv Contents

3.6.1 Maeda’s Conjecture . . . . . . . . . . . . . . . . . . . . . . 213.6.2 The Gouvea-Mazur Conjecture . . . . . . . . . . . . . . . . 22

3.7 A Modular Algorithm for Computing Characteristic Polynomials ofHecke Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.7.1 Review of Basic Facts About Modular Forms . . . . . . . . 233.7.2 The Naive Approach . . . . . . . . . . . . . . . . . . . . . . 243.7.3 The Eigenform Method . . . . . . . . . . . . . . . . . . . . 243.7.4 How to Write Down an Eigenvector over an Extension Field 263.7.5 Simple Example: Weight 36, p = 3 . . . . . . . . . . . . . . 26

4 Analytic theory of modular curves 294.1 The Modular group . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.1.1 The Upper half plane . . . . . . . . . . . . . . . . . . . . . 294.1.2 Fundamental domain for the modular group . . . . . . . . . 304.1.3 Conjugating an element of the upper half plane into the fun-

damental domain . . . . . . . . . . . . . . . . . . . . . . . . 304.1.4 Writing an element in terms of generators . . . . . . . . . . 304.1.5 Generators for the modular group . . . . . . . . . . . . . . 30

4.2 Congruence subgroups . . . . . . . . . . . . . . . . . . . . . . . . . 304.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.2.2 Fundamental domains for congruence subgroups . . . . . . 304.2.3 Coset representatives . . . . . . . . . . . . . . . . . . . . . . 304.2.4 Generators for congruence subgroups . . . . . . . . . . . . . 30

4.3 Modular curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.3.1 The upper half plane is a disk . . . . . . . . . . . . . . . . . 304.3.2 The upper half plane union the cusps . . . . . . . . . . . . 304.3.3 The Poincare metric . . . . . . . . . . . . . . . . . . . . . . 304.3.4 Fuchsian groups and Riemann surfaces . . . . . . . . . . . . 304.3.5 Riemann surfaces attached to congruence subgroups . . . . 30

4.4 Points on modular curves parameterize elliptic curves with extrastructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.5 The Genus of X(N) . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 Modular Symbols 375.1 Modular symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.2 Manin symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.2.1 Using continued fractions to obtain surjectivity . . . . . . . 395.2.2 Triangulating X(G) to obtain injectivity . . . . . . . . . . . 40

5.3 Hecke Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.4 Modular Symbols and Rational Homology . . . . . . . . . . . . . . 455.5 Special Values of L-functions . . . . . . . . . . . . . . . . . . . . . 46

6 Modular Forms of Higher Level 496.1 Modular Forms on Γ1(N) . . . . . . . . . . . . . . . . . . . . . . . 496.2 The Diamond Bracket and Hecke Operators . . . . . . . . . . . . . 50

6.2.1 Diamond Bracket Operators . . . . . . . . . . . . . . . . . . 506.2.2 Hecke Operators on q-expansions . . . . . . . . . . . . . . . 52

6.3 Old and New Subspaces . . . . . . . . . . . . . . . . . . . . . . . . 52

7 Newforms and Euler Products 55

Contents v

7.1 Atkin, Lehner, Li Theory . . . . . . . . . . . . . . . . . . . . . . . 557.2 The Up Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

7.2.1 A Connection with Galois Representations . . . . . . . . . . 607.2.2 When is Up Semisimple? . . . . . . . . . . . . . . . . . . . . 607.2.3 An Example of Non-semisimple Up . . . . . . . . . . . . . . 61

7.3 The Cusp Forms are Free of Rank One over TC . . . . . . . . . . . 617.3.1 Level 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617.3.2 General Level . . . . . . . . . . . . . . . . . . . . . . . . . . 62

7.4 Decomposing the Anemic Hecke Algebra . . . . . . . . . . . . . . . 63

8 Hecke operators as correspondences 658.1 The Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658.2 Maps induced by correspondences . . . . . . . . . . . . . . . . . . . 678.3 Induced maps on Jacobians of curves . . . . . . . . . . . . . . . . . 688.4 More on Hecke operators . . . . . . . . . . . . . . . . . . . . . . . . 698.5 Hecke operators acting on Jacobians . . . . . . . . . . . . . . . . . 69

8.5.1 The Albanese Map . . . . . . . . . . . . . . . . . . . . . . . 708.5.2 The Hecke Algebra . . . . . . . . . . . . . . . . . . . . . . . 71

8.6 The Eichler-Shimura Relation . . . . . . . . . . . . . . . . . . . . . 728.7 Applications of the Eichler-Shimura Relation . . . . . . . . . . . . 75

8.7.1 The Characteristic Polynomial of Frobenius . . . . . . . . . 758.7.2 The Cardinality of J0(N)(Fp) . . . . . . . . . . . . . . . . . 77

9 Abelian Varieties 799.1 Abelian Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 799.2 Complex Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

9.2.1 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . 809.2.2 Isogenies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 829.2.3 Endomorphisms . . . . . . . . . . . . . . . . . . . . . . . . 83

9.3 Abelian Varieties as Complex Tori . . . . . . . . . . . . . . . . . . 849.3.1 Hermitian and Riemann Forms . . . . . . . . . . . . . . . . 849.3.2 Complements, Quotients, and Semisimplicity of the Endo-

morphism Algebra . . . . . . . . . . . . . . . . . . . . . . . 859.3.3 Theta Functions . . . . . . . . . . . . . . . . . . . . . . . . 87

9.4 A Summary of Duality and Polarizations . . . . . . . . . . . . . . . 889.4.1 Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 889.4.2 The Picard Group . . . . . . . . . . . . . . . . . . . . . . . 889.4.3 The Dual as a Complex Torus . . . . . . . . . . . . . . . . 889.4.4 The Neron-Several Group and Polarizations . . . . . . . . . 899.4.5 The Dual is Functorial . . . . . . . . . . . . . . . . . . . . . 89

9.5 Jacobians of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 909.5.1 Divisors on Curves and Linear Equivalence . . . . . . . . . 909.5.2 Algebraic Definition of the Jacobian . . . . . . . . . . . . . 919.5.3 The Abel-Jacobi Theorem . . . . . . . . . . . . . . . . . . . 929.5.4 Every abelian variety is a quotient of a Jacobian . . . . . . 94

9.6 Neron Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 959.6.1 What are Neron Models? . . . . . . . . . . . . . . . . . . . 969.6.2 The Birch and Swinnerton-Dyer Conjecture and Neron Models 979.6.3 Functorial Properties of Neron Models . . . . . . . . . . . . 99

vi Contents

10 Abelian Varieties Attached to Modular Forms 10110.1 Decomposition of the Hecke Algebra . . . . . . . . . . . . . . . . . 101

10.1.1 The Dimension of Lf . . . . . . . . . . . . . . . . . . . . . . 10210.2 Decomposition of J1(N) . . . . . . . . . . . . . . . . . . . . . . . . 103

10.2.1 Aside: Intersections and Congruences . . . . . . . . . . . . 10410.3 Galois Representations Attached to Af . . . . . . . . . . . . . . . . 104

10.3.1 The Weil Pairing . . . . . . . . . . . . . . . . . . . . . . . . 10610.3.2 The Determinant . . . . . . . . . . . . . . . . . . . . . . . . 107

10.4 Remarks About the Modular Polarization . . . . . . . . . . . . . . 108

11 Modularity of Abelian Varieties 11111.1 Modularity Over Q . . . . . . . . . . . . . . . . . . . . . . . . . . . 11111.2 Modularity of Elliptic Curves over Q . . . . . . . . . . . . . . . . . 114

12 L-functions 11512.1 L-functions Attached to Modular Forms . . . . . . . . . . . . . . . 115

12.1.1 Analytic Continuation and Functional Equation . . . . . . 11612.1.2 A Conjecture About Nonvanishing of L(f, k/2) . . . . . . . 11712.1.3 Euler Products . . . . . . . . . . . . . . . . . . . . . . . . . 11812.1.4 Visualizing L-function . . . . . . . . . . . . . . . . . . . . . 119

13 The Birch and Swinnerton-Dyer Conjecture 12113.1 The Rank Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . 12113.2 Refined Rank Zero Conjecture . . . . . . . . . . . . . . . . . . . . 125

13.2.1 The Number of Real Components . . . . . . . . . . . . . . 12513.2.2 The Manin Index . . . . . . . . . . . . . . . . . . . . . . . . 12513.2.3 The Real Volume ΩA . . . . . . . . . . . . . . . . . . . . . . 12713.2.4 The Period Mapping . . . . . . . . . . . . . . . . . . . . . . 12713.2.5 Manin-Drinfeld Theorem . . . . . . . . . . . . . . . . . . . 12813.2.6 The Period Lattice . . . . . . . . . . . . . . . . . . . . . . . 12813.2.7 The Special Value L(A, 1) . . . . . . . . . . . . . . . . . . . 12813.2.8 Rationality of L(A, 1)/ΩA . . . . . . . . . . . . . . . . . . . 129

13.3 General Refined Conjecture . . . . . . . . . . . . . . . . . . . . . . 13113.4 The Conjecture for Non-Modular Abelian Varieties . . . . . . . . . 13213.5 Visibility of Shafarevich-Tate Groups . . . . . . . . . . . . . . . . . 133

13.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 13313.5.2 Every Element of H1(K,A) is Visible Somewhere . . . . . . 13413.5.3 Visibility in the Context of Modularity . . . . . . . . . . . . 13513.5.4 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . 137

References 143

Preface

This book began when the second author typed notes for the first authors 1996Berkeley course on modular forms with a view toward explaining some of the keyideas in Wiles’s celebrated proof of Fermat’s Last Theorem. The second authorthen expanded and rewrote the notes while teaching a course at Harvard in 2003on modular abelian varieties.

Kenneth A. Ribet ([email protected])William A. Stein ([email protected])

2 Contents

1The Main objects

1.1 Torsion points

The main geometric objects that we will study are elliptic curves, which are curvesof genus one curves equipped with a distinguished point. More generally, we con-sider certain algebraic curves of larger genus called modular curves, which in turngive rise via the Jacobian construction to higher-dimensional abelian varieties fromwhich we will obtain representations of the Galois group Gal(Q/Q) of the rationalnumbers.

It is convenient to view the group of complex points E(C) on an elliptic curve Eover the complex numbers C as a quotient C/L. Here

L =

∫

γ

ω∣∣ γ ∈ H1(E(C),Z)

is a lattice attached to a nonzero holomorphic differential ω on E, and the homologyH1(E(C),Z) ≈ Z×Z is the abelian group of smooth closed paths on E(C) modulothe homology relations.

Viewing E as C/L immediately gives us information about the structure of thegroup of torsion points on E, which we exploit in the next section to constructtwo-dimensional representations of Gal(Q/Q).

1.1.1 The Tate module

In the 1940s, Andre Weil studied the analogous situation for elliptic curves definedover a finite field k. He desperately wanted to find an algebraic way to describethe above relationship between elliptic curves and lattices. He found an algebraicdefinition of L/nL, when n is prime to the characteristic of k.

Let

E[n] := P ∈ E(k) : nP = 0.

4 1. The Main objects

When E is defined over C,

E[n] =

(1

nL

)/L ∼= L/nL ≈ (Z/nZ)× (Z/nZ),

so E[n] is a purely algebraic object canonically isomorphic to L/nL.For any prime `, let

E[`∞] := P ∈ E(k) : `νP = 0, some ν ≥ 1

=

∞⋃

ν=1

E[`ν ] = lim−→E[`ν ].

In an analogous way Tate constructed a rank 2 free Z`-module

T`(E) := lim←−E[`ν ],

where the map from E[`ν ] → E[`ν−1] is multiplication by `. The Z/`νZ-module

structure of E[`ν ] is compatible with the maps E[`ν ]`−→ E[`ν−1] (see, e.g., [Sil92,

III.7]), so T`(E) is free of rank 2 over Z`, and

V`(E) := T`(E)⊗Q`

is a two dimensional vector space over Q`.

1.2 Galois representations

Number theory is largely concerned with the Galois group Gal(Q/Q), which isoften studied by considering continuous linear representations

ρ : Gal(Q/Q)→ GLn(K)

whereK is a field and n is a positive integer, usually 2 in this book. Artin, Shimura,Taniyama, and Tate pioneered the study of such representations.

Let E be an elliptic curve defined over the rational numbers Q. Then Gal(Q/Q)acts on the set E[n], and this action respects the group operations, so we obtain arepresentation

ρ : Gal(Q/Q)→ Aut(E[n]) ≈ GL2(Z/nZ).

Let K be the field cut out by the ker(ρ), i.e., the fixed field of ker(ρ). Then Kis a finite Galois extension of Q since E[n] is a finite set and ker(ρ) is a normalsubgroup. Since

Gal(K/Q) ∼= Gal(Q/Q)/ ker ρ ∼= Imρ ⊆ GL2(Z/nZ)

we obtain, in this way, subgroups of GL2(Z/nZ) as Galois groups.Shimura showed that if we start with the elliptic curve E defined by the equation

y2 + y = x3 − x2 then for “most” n the image of ρ is all of GL2(Z/nZ). Moregenerally, the image is “most” of GL2(Z/nZ) when E does not have complexmultiplication. (We say E has complex multiplication if its endomorphism ringover C is strictly larger than Z.)

1.3 Modular forms 5

1.3 Modular forms

Many spectacular theorems and deep conjectures link Galois representations withmodular forms. Modular forms are extremely symmetric analytic objects, whichwe will first view as holomorphic functions on the complex upper half plane thatbehave well with respect to certain groups of transformations.

Let SL2(Z) be the group of 2× 2 integer matrices with determinant 1. For anypositive integer N , consider the subgroup

Γ1(N) :=

(a bc d

)∈ SL2(Z) : a ≡ d ≡ 1, c ≡ 0 (mod N)

of matrices in SL2(Z) that are of the form(a bc d

)1∗01 when reduced modulo N .

The space Sk(N) of cusp forms of weight k and level N for Γ1(N) consists ofall holomorphic functions f(z) on the complex upper half plane

h = z ∈ C : Im(z) > 0

that vanish at the cusps (see below) and satisfy the equation

f

(az + b

cz + d

)= (cz + d)kf(z) for all

(a bc d

)∈ Γ1(N) and z ∈ h.

Thus f(z + 1) = f(z), so f determines a function F of q(z) = e2πiz such thatF (q) = f(z). Viewing F as a function on z : 0 < |z| < 1, the condition that f(z)vanishes at infinity is that F (z) extends to a holomorphic function on z : |z| < 1and F (0) = 0. In this case, f is determined by its Fourier expansion

f(q) =∞∑

n=1

anqn.

It is also useful to consider the space Mk(N) of modular forms of level N , whichis defined in the same way as Sk(N), except that the condition that F (0) = 0 isrelaxed, and we require only that F extends to a holomorphic function at 0.

We will see in that Mk(N) and Sk(N) are finite dimensional. For example, (see) the space S12(1) has dimension one and is spanned by the famous cusp form

∆ = q∞∏

n=1

(1− qn)24 =∞∑

n=1

τ(n)qn.

The coefficients τ(n) define the Ramanujan τ -function . A non-obvious fact, whichwe will prove later using Hecke operators, is that τ is multiplicative and for everyprime p and positive integer ν, we have

τ(pν+1) = τ(p)τ(pν)− p11τ(pν−1).

1.4 Hecke operators

Mordell defined operators Tn, n ≥ 1, on Sk(N) which are called Hecke opera-tors. These proved very fruitful. The set of such operators forms a commuting

6 1. The Main objects

family of endomorphisms and is hence “almost” simultaneously diagonalizable.The precise meaning of “almost” and the actual structure of the Hecke algebraT = Q[T1, T2, . . .] will be studied in greater detail in.

Often there is a basis f1, . . . , fr of Sk(N) such that each f = fi =∑∞n=1 anq

n is asimultaneous eigenvector for all the Hecke operators Tn and, moreover, Tnf = anf .In this situation, the eigenvalues an are necessarily algebraic integers and the fieldQ(. . . , an, . . .) generated by all an is finite over Q (see ).

The an exhibit remarkable properties. For example,

τ(n) ≡∑

d|nd11 (mod 691),

as we will see in . The key to studying and interpreting the an is to understandthe deep connections between Galois representations and modular forms that werediscovered by Serre, Shimura, Eichler and Deligne.

2Modular representations and algebraiccurves

2.1 Arithmetic of modular forms

Let us give ourselves a cusp form

f =

∞∑

n=1

anqn ∈ Sk(N)

which is an eigenform for all of the Hecke operators Tp. Then the Mellin trans-form of f is the L-function

L(f, s) =

∞∑

n=1

anns.

Let K = Q(a1, a2, . . .). One can show that the an are algebraic integers and thatK is a number field. When k = 2 Shimura associated to f an abelian variety Afover Q of dimension [K : Q] on which Z[a1, a2, . . .] acts [Shi94, Theorem 7.14].

Example 2.1.1 (Modular Elliptic Curves). Suppose now that all coefficients an off lie in Q so that [K : Q] = 1 and hence Af is a one dimensional abelian variety.A one dimensional abelian variety is an elliptic curve. An elliptic curve isogenousto one arising via this construction is called modular.

Elliptic curves E1 and E2 are isogenous if there is a morphism E1 → E2 ofalgebraic groups, having finite kernel.

The following “modularity conjecture” motivates much of the theory discussedin this course. It is now a theorem of Breuil, Conrad, Diamond, Taylor, and Wiles(see []).

Conjecture 2.1.2 (Shimura-Taniyama). Every elliptic curve over Q is modu-lar, that is, isogenous to a curve constructed in the above way.

For k ≥ 2 Serre and Deligne discovered a way to associate to f a family of `-adicrepresentations. Let ` be a prime number and K = Q(a1, a2, . . .) be as above.

8 2. Modular representations and algebraic curves

Then it is well known that

K ⊗Q Q`∼=∏

λ|`Kλ.

One can associate to f a representation

ρ`,f : G = Gal(Q/Q)→ GL(K ⊗Q Q`)

unramified at all primes p - `N . For ρ`,f to be unramified we mean that forall primes P lying over p, the inertia group of the decomposition group at P iscontained in the kernel of ρ`,f . The decomposition groupDP at P is the set of thoseg ∈ G which fix P . Let k be the residue fieldO/P whereO is the ring of all algebraicintegers. Then the inertia group IP is the kernel of the map DP → Gal(k/k).

Now IP ⊂ DP ⊂ Gal(Q/Q) and DP /IP is cyclic (being isomorphic to a sub-group of the Galois group of a finite extension of finite fields) so it is generated bya Frobenious automorphism Frobp lying over p. One has

tr(ρ`,f (Frobp)) = ap ∈ K ⊂ K ⊗Q`

and

det(ρ`,f ) = χk−1` ε

where χ` is the `th cyclotomic character and ε is the Dirichlet character asso-ciated to f . There is an incredible amount of “abuse of notation” packed intothis statement. First, the Frobenius FrobP (note P not p) is only well defined inGal(K/Q) (so I think an unstated result is that K must be Galois), and then Frobpis only well defined up to conjugacy. But this works out since ρ`,f is well-defined onGal(K/Q) (it kills Gal(Q/K)) and the trace is well-defined on conjugacy classes(tr(AB) = tr(BA) so tr(ABA−1) = Tr(B)).

2.2 Characters

Let f ∈ Sk(N), then for all(a bc d

)∈ 2z with c ≡ 0 mod N we have

f(az + b

cz + d) = (cz + d)kε(d)f(z)

where ε : (Z/NZ)∗ → C∗ is a Dirichlet character mod N . If f is an eigenform forthe so called “diamond-bracket operator” 〈d〉 so that f |〈d〉 = ε(d)f then ε actuallytakes values in K.

Led ϕN be the mod N cyclotomic character so that ϕN : G → (Z/NZ)∗ takesg ∈ G to the automorphism induced by g on the Nth cyclotomic extension Q(µµµN )of Q (where we identify Gal(Q(µµµN )/Q) with (Z/NZ)∗). Then what we called εabove in the formula det(ρ`) = χk−1

` ε is really the composition

GϕN−−→ (Z/NZ)∗

ε−→ C∗.

For each positive integer ν we consider the `νth cyclotomic character on G,

ϕ`ν : G→ (Z/`νZ)∗.

Putting these together gives the `-adic cyclotomic character

χ` : G→ Z∗` .

2.3 Parity Conditions 9

2.3 Parity Conditions

Let c ∈ Gal(Q/Q) be complex conjugation. Then ϕN (c) = −1 so ε(c) = ε(−1)and χk−1

` (c) = (−1)k−1. Now let(a bc d

)=(−1 0

0 −1

), then for f ∈ Sk(N),

f(z) = (−1)kε(−1)f(z)

so (−1)kε(−1) = 1 thus

det(ρ`,f (c)) = ε(−1)(−1)k−1 = −1.

Thus the det character is odd so the representation ρ`,f is odd.

Remark 2.3.1 (Vague Question). How can one recognize representations like ρ`,f“in nature”? Mazur and Fontaine have made relevant conjectures. The Shimura-Taniyama conjecture can be reformulated by saying that for any representationρ`,E comming from an elliptic curve E there is f so that ρ`,E ∼= ρ`,f .

2.4 Conjectures of Serre (mod ` version)

Suppose f is a modular form, ` ∈ Z prime, λ a prime lying over `, and therepresentation

ρλ,f : G→ GL2(Kλ)

(constructed by Serre-Deligne) is irreducible. Then ρλ,f is conjugate to a represen-tation with image in GL2(Oλ), where Oλ is the ring of integers of Kλ. Reducingmod λ gives a representation

ρλ,f : G→ GL2(Fλ)

which has a well-defined trace and det, i.e., the det and trace don’t depend on thechoice of conjugate representation used to obtain the reduced representation. Oneknows from representation theory that if such a representation is semisimple thenit is completely determined by its trace and det (more precisely, the characteristicpolynomials of all of its elements – see chapter ??). Thus if ρλ,f is irreducible (andhence semisimple) then it is unique in the sense that it does not depend on thechoice of conjugate.

2.5 General remarks on mod p Galois representations

[[This section was written by Joseph Loebach Wetherell.]]First, what are semi-simple and irreducible representations? Remember that a

representation ρ is a map from a group G to the endomorphisms of some vectorspace W (or a free module M if we are working over a ring instead of a field, butlet’s not worry about that for now). A subspace W ′ of W is said to be invariantunder ρ if ρ takes W ′ back into itself. (The point is that if W ′ is invariant, then ρinduces representations on both W ′ and W/W ′.) An irreducible representation isone where the only invariant subspaces are 0 and W . A semi-simple representationis one where for every invariant subspace W ′ there is a complementary invariantsubspace W ′′ – that is, you can write ρ as the direct sum of ρ|W ′ and ρ|W ′′ .


Another way to say this is that if W ′ is an invariant subspace then we get ashort exact sequence

0→ ρ|W/W ′ → ρ→ ρ|W ′ → 0.

Furthermore ρ is semi-simple if and only if every such sequence splits.Note that irreducible representations are semi-simple.One other fact is that semi-simple Galois representations are uniquely deter-

mined (up to isomorphism class) by their trace and determinant.Now, since in the case we are doing, G = Gal(Q/Q) is compact, it follows that

the image of any Galois representation ρ into GL2(Kλ) is compact. Thus we canconjugate it into GL2(Oλ). Irreducibility is not needed for this.

Now that we have a representation into GL2(Oλ), we can reduce to get a repre-sentation ρ to GL2(Fλ). This reduced representation is not uniquely determinedby ρ, since we had a choice of conjugators. However, the trace and determinantare invariant under conjugation, so the trace and determinant of the reduced rep-resentation are uniquely determined by ρ.

So we know the trace and determinant of the reduced representation. If we alsoknew that it was semi-simple, then we would know its isomorphism class, and wewould be done. So we would be happy if the reduced representation is irreducible.And in fact, it is easy to see that if the reduced representation is irreducible, thenρ must also be irreducible. Now, it turns out that all ρ of interest to us will beirreducible; unfortunately, we can’t go the other way and claim that ρ irreducibleimplies the reduction is irreducible.

2.6 Serre’s Conjecture

Serre has made the following conjecture which is still open at the time of thiswriting.

Conjecture 2.6.1 (Serre). All irreducible representation of G over a finite fieldwhich are odd, i.e., det(σ(c)) = −1, c complex conjugation, are of the form ρλ,ffor some representation ρλ,f constructed as above.

Example 2.6.2. Let E/Q be an elliptic curve and let σ` : G → GL2(F`) be therepresentation induced by the action of G on the `-torsion of E. Then detσ` = ϕ`is odd and σ` is usually irreducible, so Serre’s conjecture would imply that σ` ismodular. From this one can, assuming Serre’s conjecture, prove that E is modular.

Let σ : G→ GL2(F) (F is a finite field) be a represenation of the Galois groupG. The we say that the representions σ is modular if there is a modular form f , aprime λ, and an embedding F → Fλ such that σ ∼= ρλ,f over Fλ.

For more details, see Chapter ?? and [RS01].

2.7 Wiles’s Perspective

Suppose E/Q is an elliptic curve and ρ`,E : G → GL2(Z`) the associated `-adic representation on the Tate module T`. Then by reducing we obtain a mod `representation

ρ`,E = σ`,E : G→ GL2(F`).

2.7 Wiles’s Perspective 11

If we can show this representation is modular for infinitely many ` then we willknow that E is modular.

Theorem 2.7.1 (Langland’s and Tunnel). If σ2,E and σ3,E are irreducible,then they are modular.

This is proved by using that GL2(F2) and GL2(F3) are solvable so we may apply“base-change”.

Theorem 2.7.2 (Wiles). If ρ is an `-adic representation which is irreducible andmodular mod ` with ` > 2 and certain other reasonable hypothesis are satisfied, thenρ itself is modular.

3Modular Forms of Level 1

In this chapter, we view modular forms of level 1 both as holomorphic functionson the upper half plane and functions on lattices. We then define Hecke operatorson modular forms, and derive explicit formulas for the action of Hecke operatorson q-expansions. An excellent reference for the theory of modular forms of level 1is Serre [Ser73, Ch. 7].

3.1 The Definition

Let k be an integer. The space Sk = Sk(1) of cusp forms of level 1 and weight kconsists of all functions f that are holomorphic on the upper half plane h and suchthat for all

(a bc d

)∈ SL2(Z) one has

f

(aτ + b

cτ + d

)= (cτ + d)kf(τ), (3.1.1)

and f vanishes at infinity, in a sense which we will now make precise. The matrix( 1 1

0 1 ) is in SL2(Z), so f(τ + 1) = f(τ). Thus f passes to a well-defined functionof q(τ) = e2πiτ . Since for τ ∈ h we have |q(τ)| < 1, we may view f = f(q) as afunction of q on the punctured open unit disc q : 0 < |q| < 1. The conditionthat f(τ) vanishes at infinity means that f(q) extends to a holomorphic functionon the open disc z : |z| < 1 so that f(0) = 0. Because holomorphic functions arerepresented by power series, there is a neighborhood of 0 such that

f(q) =

∞∑

n=1

anqn,

so for all τ ∈ h with sufficiently large imaginary part, f(τ) =∑∞n=1 ane

2πinτ .It will also be useful to consider the slightly large space Mk(1) of holomorphic

functions on h that transform as above and are merely required to be holomorphicat infinity.

14 3. Modular Forms of Level 1

Remark 3.1.1. In fact, the series∑∞n=1 ane

2πinτ converges for all τ ∈ h. This isbecause the Fourier coefficients an are O(nk/2) (see [Miy89, Cor. 2.1.6, pg. 43]).

Remark 3.1.2. In [Ser73, Ch. 7], the weight is defined in the same way, but in thenotation our k is twice his k.

3.2 Some Examples and Conjectures

The space Sk(1) of cusp forms is a finite-dimensional complex vector space. For keven we have dimSk(1) = bk/12c if k 6≡ 2 (mod 12) and bk/12c − 1 if k ≡ 2(mod 12), except when k = 2 in which case the dimension is 0. For even k, thespace Mk(1) has dimension 1 more than the dimension of Sk(1), except when k = 2when both have dimension 0. (For proofs, see, e.g., [Ser73, Ch. 7, §3].)

By the dimension formula mentioned above, the first interesting example is thespace S12(1), which is a 1-dimensional space spanned by

∆(q) = q

∞∏

n=1

(1− qn)24

= q − 24q2 + 252q3 − 1472q4 + 4830q5 − 6048q6 − 16744q7 + 84480q8

− 113643q9 − 115920q10 + 534612q11 − 370944q12 − 577738q13 + · · ·

That ∆ lies in S12(1) is proved in [Ser73, Ch. 7, §4.4] by expressing ∆ in termsof elements of M4(1) and M6(1), and computing the q-expansion of the resultingexpression.

The Ramanujan τ function τ(n) assigns to n the nth coefficient of ∆(q).

Conjecture 3.2.1 (Lehmer). τ(n) 6= 0 for all n ≥ 1.

This conjecture has been verified for n ≤ 22689242781695999 (see Jordan andKelly, 1999).

Conjecture 3.2.2 (Edixhoven). Let p be a prime. There a polynomial timealgorithm to compute τ(p), polynomial in the number of digits of p.

Edixhoven has proposed an approach to find such an algorithm. His idea is touse `-adic cohomology to find an analogue of the Schoof-Elkies-Atkin algorithm(which counts the number Nq of points on an elliptic curves over a finite field Fqby computing Nq mod ` for many primes `). Here’s what Edixhoven has to sayabout the status of his conjecture (email, October 22, 2003):

I have made a lot of progress on proving that my method runs inpolynomial time, but it is not yet complete. I expect that all shouldbe completed in 2004. For higher weights [...] you need to compute onvarying curves such as X1(`) for ` up to log(p) say.

An important by-product of my method is the computation of themod ` Galois representations associated to ∆ in time polynomial in `.So, it should be seen as an attempt to make the Langlands correspon-dence for GL2 over Q available computationally.

If f ∈ Mk(1) and g ∈ Mk′(1), then it is easy to see from the definitions thatfg ∈ Mk+k′(1). Moreover, ⊕k≥0Mk(1) is a commutative graded ring generated

3.3 Modular Forms as Functions on Lattices 15

freely by E4 = 1 + 240∑∞n=1 σ3(n)qn and E6 = 1 − 504

∑∞n=1 σ5(n)qn, where

σd(n) is the sum of the dth powers of the positive divisors of n (see [Ser73, Ch.7,§3.2]).Example 3.2.3. Because E4 and E6 generate, it is straightforward to write downa basis for any space Mk(1). For example, the space M36(1) has basis

f1 = 1 + 6218175600q4 + 15281788354560q5 + · · ·f2 = q + 57093088q4 + 37927345230q5 + · · ·f3 = q2 + 194184q4 + 7442432q5 + · · ·f4 = q3 − 72q4 + 2484q5 + · · ·

3.3 Modular Forms as Functions on Lattices

In order to define Hecke operators, it will be useful to view modular forms asfunctions on lattices in C.

A lattice L ⊂ C is a subring L = Zω1 + Zω2 for which ω1, ω2 ∈ C are linearlyindependent over R. We may assume that ω1/ω2 ∈ h = z ∈ C : Im(z) > 0. LetR be the set of all lattices in C. Let E be the set of isomorphism classes of pairs(E,ω), where E is an elliptic curve over C and ω ∈ Ω1

E is a nonzero holomorphicdifferential 1-form on E. Two pairs (E,ω) and (E ′, ω′) are isomorphic if there isan isomorphism ϕ : E → E′ such that ϕ∗(ω′) = ω.

Proposition 3.3.1. There is a bijection between R and E under which L ∈ Rcorresponds to (C/L, dz) ∈ E.

Proof. We describe the maps in each direction, but leave the proof that theyinduce a well-defined bijection as an exercise for the reader. Given L ∈ R, byWeierstrass theory the quotient C/L is an elliptic curve, which is equipped withthe distinguished differential ω induced by the differential dz on C.

Conversely, if E is an elliptic curve over C and ω ∈ Ω1E is a nonzero differential,

we obtain a lattice L in C by integrating homology classes:

L = Lω =

∫

γ

ω : γ ∈ H1(E(C),Z)

.

LetB = (ω1, ω2) : ω1, ω2 ∈ C, ω1/ω2 ∈ h ,

be the set of ordered basis of lattices in C, ordered so that ω1/ω2 ∈ h. There is aleft action of SL2(Z) on B given by

(a bc d

)(ω1, ω2) 7→ (aω1 + bω2, cω1 + dω2)

and SL2(Z)\B ∼= R. (The action is just the left action of matrices on columnvectors, except we write (ω1, ω2) as a row vector since it takes less space.)

Give a modular form f ∈Mk(1), associate to f a function F : R → C as follows.First, on lattices of the special form Zτ + Z, for τ ∈ h, let F (Zτ + Z) = f(τ).


In order to extend F to a function on all lattices, suppose further that F satisfiesthe homogeneity condition F (λL) = λ−kF (L), for any λ ∈ C and L ∈ R. Then

F (Zω1 + Zω2) = ω−k2 F (Zω1/ω2 + Z) := ω−k

2 f(ω1/ω2).

That F is well-defined exactly amounts to the transformation condition (3.1.1)that f satisfies.

Lemma 3.3.2. The lattice function F : R → C associated to f ∈ Mk(1) is welldefined.

Proof. Suppose Zω1 + Zω2 = Zω′1 + Zω′

2 with ω1/ω2 and ω′1/ω

′2 both in h. We

must verify that ω−k2 f(ω1/ω2) = (ω′

2)−kf(ω′

1/ω′2). There exists

(a bc d

)∈ SL2(Z)

such that ω′1 = aω1 + bω2 and ω′

2 = cω1 + dω2. Dividing, we see that ω′1/ω

′2 =(

a bc d

)(ω1/ω2). Because f is a weight k modular form, we have

f

(ω′

1

ω′2

)= f

((a bc d

)(ω1

ω2

))=

(cω1

ω2+ d

)kf

(ω1

ω2

).

Multiplying both sides by ωk2 yields

ωk2f

(ω′

1

ω′2

)= (cω1 + dω2)

kf

(ω1

ω2

).

Observing that ω′2 = cω1 + dω2 and dividing again completes the proof.

Since f(τ) = F (Zτ+Z), we can recover f from F , so the map f 7→ F is injective.Moreover, it is surjective in the sense that if F is homogeneous of degree −k, thenF arises from a function f : h → C that transforms like a modular form. Moreprecisely, if F : R → C satisfies the homogeneity condition F (λL) = λ−kF (L),then the function f : h→ C defined by f(τ) = F (Zτ+Z) transforms like a modularform of weight k, as the following computation shows: For any

(a bc d

)∈ SL2(Z) and

τ ∈ h, we have

f

(aτ + b

cτ + d

)= F

(Zaτ + b

cτ + d+ Z

)

= F ((cτ + d)−1 (Z(aτ + b) + Z(cτ + d)))

= (cτ + d)kF (Z(aτ + b) + Z(cτ + d))

= (cτ + d)kF (Zτ + Z)

= (cτ + d)kf(τ).

Say that a function F : R → C is holomorphic on h ∪ ∞ if the function f(τ) =F (Zτ + Z) is. We summarize the above discussion in a proposition.

Proposition 3.3.3. There is a bijection between Mk(1) and functions F : R →C that are homogeneous of degree −k and holomorphic on h ∪ ∞. Under thisbijection F : R → C corresponds to f(τ) = F (Zτ + Z).

3.4 Hecke Operators 17

3.4 Hecke Operators

Define a map Tn from the free abelian group generated by all C-lattices into itselfby

Tn(L) =∑

L′⊂L[L:L′]=n

L′,

where the sum is over all sublattices L′ ⊂ L of index n. For any function F : R → Con lattices, define Tn(F ) : R → C by

(Tn(F ))(L) = nk−1∑

L′⊂L[L:L′]=n

F (L′).

Note that if F is homogeneous of degree −k, then Tn(F ) is also homogeneousof degree −k.

Since (n,m) = 1 implies TnTm = Tnm and Tpk is a polynomial in Z[Tp] (see[Ser73, Cor. 1, pg. 99]), the essential case to consider is n prime.

Suppose L′ ⊂ L with [L : L′] = n. Then every element of L/L′ has orderdividing n, so nL ⊂ L′ ⊂ L and

L′/nL ⊂ L/nL ≈ (Z/nZ)2.

Thus the subgroups of (Z/nZ)2 of order n correspond to the sublattices L′ of L ofindex n. When n = ` is prime, there are `+1 such subgroups, since the subgroupscorrespond to nonzero vectors in F` modulo scalar equivalence, and there are(`2 − 1)(`− 1) = `+ 1 of them.

Recall from Proposition 3.3.1 that there is a bijection between the set R oflattices in C and the set E of isomorphism classes of pairs (E,ω), where ω is anonzero differential on E.

Suppose F : R → C is homogeneous of degree −k, so F (λL) = λ−kF (L). Thenwe may also view T` as a sum over lattices that contain L with index `, as follows.Suppose L′ ⊂ L is a sublattice of index ` and set L′′ = `−1L′. Then we have achain of inclusions

`L ⊂ L′ ⊂ L ⊂ `−1L′ = L′′.

Since [`−1L′ : L′] = `2 and [L : L′] = `, it follows that [L′′ : L] = `. By homogeneity,

T`(F )(L) = `k−1∑

[L:L′]=`

F (L′) =1

`

∑

[L′′:L]=`

F (L′′). (3.4.1)

3.5 Hecke Operators Directly on q-expansions

Recall that the nth Hecke operator Tn of weight k is

Tn(L) = nk−1∑

L′⊂L[L:L′]=n

L′.

Modular forms of weight k correspond to holomorphic functions on lattices ofdegree −k, and Tn extend to an operator on these functions on lattices, so Tn


defines on operator on Mk(1). Recall that Fourier expansion defines an injectivemap Mk(1) ⊂ C[[q]]. In this section, we describe Tn(

∑anq

n) explicitly as a q-expansion.

3.5.1 Explicit Description of Sublattices

In order to describe Tn more explicitly, we explicitly enumerate the sublatticesL′ ⊂ L of index n. More precisely, we give a basis for each L′ in terms of a basisfor L. Note that L/L′ is a group of order n and

L′/nL ⊂ L/nL = (Z/nZ)2.

Write L = Zω1 + Zω2, let Y2 be the cyclic subgroup of L/L′ generated by ω2 andlet d = #Y2. If Y1 = (L/L′)/Y2, then Y1 is generated by the image of ω1, so it is acyclic group of order a = n/d. Our goal is to exhibit a basis of L′. Let ω′

2 = dω2 ∈ L′

and use that Y1 is generated by the image of ω1 to write aω1 = ω′1 − bω2 for some

integer b and some ω′1 ∈ L′. Since b is only well-defined modulo d we may assume

0 ≤ b ≤ d− 1. Thus (ω′1

ω′2

)=(a b0 d

)(ω1

ω2

)

and the change of basis matrix has determinant ad = n. Since

Zω′1 + Zω′

2 ⊂ L′ ⊂ L = Zω1 + Zω2

and [L : Zω′1 +Zω′

2] = n (since the change of basis matrix has determinant n) and[L : L′] = n we see that L′ = Zω′

1 + Zω′2.

Proposition 3.5.1. Let n be a positive integer. There is a one-to-one correspon-dence between sublattices L′ ⊂ L of index n and matrices

(a b0 d

)with ad = n and

0 ≤ b ≤ d− 1.

Proof. The correspondence is described above. To check that it is a bijection, wejust need to show that if γ =

(a b0 d

)and γ′ =

(a′ b′

0 d′

)are two matrices satisfying

the listed conditions, and

Z(aω1 + bω2) + Zdω2 = Z(aω′1 + bω′

2) + Zdω′2,

then γ = γ′. Equivalently, if σ ∈ SL2(Z) and σγ = γ′, then σ = 1. To see this, wecompute

σ = γ′γ−1 =1

n

(a′d ab′ − a′b

0 ad′

).

Since σ ∈ SL2(Z), we have n | a′d, and n | ad′, and aa′dd′ = n2. If a′d > n, thenbecause aa′dd′ = n2, we would have ad′ < n, which would contradict the factthat n | ad′; also, a′d < n is impossible since n | a′d. Thus a′d = n and likewisead′ = n. Since ad = n as well, it follows that a′ = a and d′ = d, so σ = ( 1 t

0 1 ) forsome t ∈ Z. Then σγ =

(a b+dt0 d

), which implies that t = 0, since 0 ≤ b ≤ d − 1

and 0 ≤ b+ dt ≤ d− 1.

Remark 3.5.2. As mentioned earlier, when n = ` is prime, there are `+1 sublatticesof index `. In general, the number of such sublattices is the sum of the positivedivisors of n (exercise).

3.5 Hecke Operators Directly on q-expansions 19

3.5.2 Hecke operators on q-expansions

Recall that if f ∈Mk(1), then f is a holomorphic functions on h ∪ ∞ such that

f(τ) = f

(aτ + b

cτ + d

)(cτ + d)−k

for all(a bc d

)∈ SL2(Z). Using Fourier expansion we write

f(τ) =

∞∑

m=0

cme2πiτm,

and say f is a cusp form if c0 = 0. Also, there is a bijection between modularforms f of weight k and holomorphic lattice functions F : R → C that satisfyF (λL) = λ−kF (L); under this bijection F corresponds to f(τ) = F (Zτ + Z).

Now assume f(τ) =∑∞m=0 cmq

m is a modular form with corresponding latticefunction F . Using the explicit description of sublattices from Section 3.5.1 above,we can describe the action of the Hecke operator Tn on the Fourier expansion off(τ), as follows:

TnF (Zτ + Z) = nk−1∑

a,b,dab=n

0≤b≤d−1

F ((aτ + b)Z + dZ)

= nk−1∑

d−kF

(aτ + b

dZ + Z

)

= nk−1∑

d−kf

(aτ + b

d

)

= nk−1∑

a,d,b,m

d−kcme2πi( aτ+b

d )m

= nk−1∑

a,d,m

d1−kcme2πiamτ

d1

d

d−1∑

b=0

(e

2πimd

)b

= nk−1∑

ad=nm′≥0

d1−kcdm′e2πiam′τ

=∑

ad=nm′≥0

ak−1cdm′qam′

.

In the second to the last expression we let m = dm′ for m′ ≥ 0, then used thatthe sum 1

d

∑d−1b=0 (e

2πimd )b is only nonzero if d | m.

ThusTnf(q) =

∑

ad=nm≥0

ak−1cdmqam

and if µ ≥ 0 then the coefficient of qµ is∑

a|na|µ

ak−1cnµ

a2.

(To see this, let m = a/µ and d = n/a and substitute into the formula above.)


Remark 3.5.3. When k ≥ 1 the coefficients of qµ for all µ belong to the Z-modulegenerated by the cm.

Remark 3.5.4. Setting µ = 0 gives the constant coefficient of Tnf which is

∑

a|nak−1c0 = σk−1(n)c0.

Thus if f is a cusp form so is Tnf . (Tnf is holomorphic since its original definitionis as a finite sum of holomorphic functions.)

Remark 3.5.5. Setting µ = 1 shows that the coefficient of q in Tnf is∑a|1 1k−1cn =

cn. As an immediate corollary we have the following important result.

Corollary 3.5.6. If f is a cusp form such that Tnf has 0 as coefficient of q forall n ≥ 1, then f = 0.

When n = p is prime, the action action of Tp on the q-expansion of f is givenby the following formula:

Tpf =∑

µ≥0

∑

a|na|µ

ak−1cnµ

a2qµ.

Since n = p is prime, either a = 1 or a = p. When a = 1, cpµ occurs in thecoefficient of qµ and when a = p, we can write µ = pλ and we get terms pk−1cλ inqλp. Thus

Tpf =∑

µ≥0

cpµqµ + pk−1

∑

λ≥0

cλqpλ.

3.5.3 The Hecke Algebra and Eigenforms

Definition 3.5.7 (Hecke Algebra). The Hecke algebra T associated to Mk(1)is the subring of End(Mk(1)) generated by the operators Tn for all n. Similarly,the Hecke algebra associated to Sk(1) is the subring of End(Sk(1)) generated byall Hecke operators Tn.

The Hecke algebra is commutative (e.g., when (n,m) = 1 we have TnTm =Tnm = Tmn = TmTn) of finite rank over Z.

Definition 3.5.8 (Eigenform). An eigenform f ∈ Mk(1) is a nonzero elementsuch that f is an eigenvector for every Hecke operator Tn. If f ∈ Sk(1) is aneigenform, then f is normalized if the coefficient of q in the q-expansion of f is 1.We sometimes called a normalized cuspidal eigenform a newform.

If f =∑∞n=1 cnq

n is a normalized eigenform, then Remark 3.5.5 implies thatTn(f) = cnf . Thus the coefficients of a newform are exactly the system of eigen-values of the Hecke operators acting on the newform.

Remark 3.5.9. It follows from Victor Miller’s thesis that T1, . . . , Tn generate T ⊂Sk(1), where n = dimSk(1).

3.5.4 Examples

> M := ModularForms(1,12);

3.6 Two Conjectures about Hecke Operators on Level 1 Modular Forms 21

> HeckeOperator(M,2);

[ 2049 196560]

[ 0 -24]

> S := CuspidalSubspace(M);

> HeckeOperator(S,2);

[-24]

> Factorization(CharacteristicPolynomial(HeckeOperator(M,2)));

[

<x - 2049, 1>,

<x + 24, 1>

]

> M := ModularForms(1,40);

> M;

Space of modular forms on Gamma_0(1) of weight 40 and dimension 4

over Integer Ring.

> Basis(M);

[

1 + 1250172000*q^4 + 7541401190400*q^5 + 9236514405888000*q^6

+ 3770797689077760000*q^7 + O(q^8),

q + 19291168*q^4 + 37956369150*q^5 + 14446985236992*q^6 +

1741415886056000*q^7 + O(q^8),

q^2 + 156024*q^4 + 57085952*q^5 + 1914094476*q^6 -

27480047616*q^7 + O(q^8),

q^3 + 168*q^4 - 12636*q^5 + 392832*q^6 - 7335174*q^7 + O(q^8)

]

> HeckeOperator(M,2);

[549755813889 0 1250172000 9236514405888000]

[0 0 549775105056 14446985236992]

[0 1 156024 1914094476]

[0 0 168 392832]

> Factorization(CharacteristicPolynomial(HeckeOperator(M,2)));

[

<x - 549755813889, 1>,

<x^3 - 548856*x^2 - 810051757056*x + 213542160549543936, 1>

]

3.6 Two Conjectures about Hecke Operators on Level 1Modular Forms

3.6.1 Maeda’s Conjecture

Conjecture 3.6.1 (Maeda). Let k be a positive integer such that Sk(1) haspositive dimension and let T ⊂ End(Sk(1)) be the Hecke algebra. Then there isonly one Gal(Q/Q) orbit of normalized eigenforms of level 1.

There is some numerical evidence for this conjecture. It is true for k ≤ 2000,according to [FJ02]. Buzzard shows in [Buz96] that for the weights k ≤ 228 with


k/12 a prime, the Galois group of the characteristic polynomial of T2 is the fullsymmetric group.

Possible student project: I have computed the characteristic polynomial ofT2 for all weights k ≤ 3000:

http://modular.fas.harvard.edu/Tables/charpoly level1/t2/

However, I never bothered to try to prove that these are all irreducible, which wouldestablish Maeda’s conjecture for k ≤ 3000. The MathSciNet reviewer of [FJ02] said“In the present paper the authors take a big step forward towards proving Maeda’sconjecture in the affirmative by establishing that the Hecke polynomial Tp,k(x) isirreducible and has full Galois group over Q for k ≤ 2000 and p < 2000, p prime.”Thus stepping forward to k ≤ 3000, at least for p = 2, might be worth doing.

3.6.2 The Gouvea-Mazur Conjecture

Fix a prime p, and let Fp,k ∈ Z[x] be the characteristic polynomial of Tp acting onMk(1). The slopes of Fp,k are the p-adic valuations ordp(α) ∈ Q of the roots α ∈ Qp

of Fp,k. They can be computed easily using Newton polygons. For example, thep = 5 slopes for F5,12 are 0, 1, 1, for F5,12+4·5 they are 0, 1, 1, 4, 4, and for F5,12+4·52

they are 0, 1, 1, 5, 5, 5, 5, 5, 5, 10, 10, 11, 11, 14, 14, 15, 15, 16, 16.

> function s(k,p)

return NewtonSlopes(CharacteristicPolynomial(

HeckeOperator(ModularForms(1,k),p)),p);

end function;

> s(12,5);

[* 0, 1 *]

> s(12+4*5,5);

[* 0, 1, 4 *]

> s(12+4*5^2,5);

[* 0, 1, 5, 5, 5, 10, 11, 14, 15, 16 *]

> s(12+4*5^3,5);

[* 0, 1, 5, 5, 5, 10, 11, 14, 15, 16, 20, 21, 24, 25, 27, 30, 31,

34, 36, 37, 40, 41, 45, 46, 47, 50, 51, 55, 55, 55, 59, 60, 63,

64, 65, 69, 70, 73, 74, 76, 79, 80, 83 *]

Let d(k, α, p) be the multiplicity of α as a slope of Fp,k.

Conjecture 3.6.2 (Gouvea-Mazur, 1992). Fix a prime p and a nonnegativerational number α. Suppose k1 and k2 are integers with k1, k2 ≥ 2α+2, and k1 ≡ k2

(mod pn(p− 1)) for some n ≥ α. Then d(k1, α, p) = d(k2, α, p).

Notice that the above examples, with p = 5 and k1 = 12, are consistent withthis conjecture. However, the conjecture is false in general. Frank Calegari andKevin Buzzard recently found the first counterexample, when p = 59, k1 = 16,α = 1, and k2 = 16 + 59 · 58 = 3438. We have d(16, 0, 59) = 0d(16, 1, 59) = 1,d(16, α, 59) = 0 for all other α. However, initial computations strongly suggest(but do not prove!) that d(3438, 1, 59) = 2. It is a finite, but difficult, computationto decide what d(3438, 1, 59) really is (see Section 3.7). Using a trace formula,Calegari and Buzzard at least showed that either d(3438, 1, 59) ≥ 2 or there existsα < 1 such that d(3438, α, 59) > 0, both of which contradict Conjecture 3.6.2.

3.7 A Modular Algorithm for Computing Characteristic Polynomials of Hecke Operators 23

There are many theorems about more general formulations of the Gouvea-Mazurconjecture, and a whole geometric theory “the Eigencurve” [CM98] that helpsexplain it, but discussing this further is beyond the scope of this book.

3.7 A Modular Algorithm for Computing CharacteristicPolynomials of Hecke Operators

In computational investigations, it is frequently useful to compute the charac-teristic polynomial Tp,k of the Hecke operator Tp acting on Sk(1). This can beaccomplished in several ways, each of which has advantages. The Eichler-Selbergtrace formula (see Zagier’s appendix to [Lan95, Ch. III]), can be used to computethe trace of Tn,k, for n = 1, p, p2, . . . , pd−1, where d = dimSk(1), and from thesetraces it is straightforward to recover the characteristic polynomial of Tp,k. Usingthe trace formula, the time required to compute Tr(Tn,k) grows “very quickly”in n (though not in k), so this method becomes unsuitable when the dimensionis large or p is large, since pd−1 is huge. Another alternative is to use modularsymbols of weight k, as in [Mer94], but if one is only interested in characteristicpolynomials, little is gained over more naive methods (modular symbols are mostuseful for investigating special values of L-functions).

In this section, we describe an algorithm to compute the characteristic polyno-mial of the Hecke operator Tp,k, which is adapted for the case when p > 2. It couldbe generalized to modular forms for Γ1(N), given a method to compute a basisof q-expansions to “low precision” for the space of modular forms of weight k andlevel N . By “low precision” we mean to precision O(qdp+1), where T1, T2, . . . , Tdgenerate the Hecke algebra T as a ring. The algorithm described here uses nothingmore than the basics of modular forms and some linear algebra; in particular, notrace formulas or modular symbols are involved.

3.7.1 Review of Basic Facts About Modular Forms

We briefly recall the background for this section. Fix an even integer k. Let Mk(1)denote the space of weight k modular forms for SL2(Z) and Sk(1) the subspace ofcusp forms. Thus Mk(1) is a C-vector space that is equipped with a ring

T = Z[. . . Tp,k . . .] ⊂ End(Mk(1))

of Hecke operators. Moreover, there is an injective q-expansion map Mk(1) →C[[q]]. For example, when k ≥ 4 there is an Eisenstein series Ek, which lies inMk(1). The first two Eisenstein series are

E4(q) =1

240+∑

n≥1

σ3(n)qn and E6(q) =1

504+∑

n≥1

σ5(n)qn,

where q = e2πiz, σk−1(n) is the sum of the k − 1st power of the positive divisors.For every prime number p, the Hecke operator Tp,k acts on Mk(1) by

Tp,k

∑

n≥0

anqn

=

∑

n≥0

anpqn + pk−1anq

np. (3.7.1)


Proposition 3.7.1. The set of modular forms Ea4Eb6 is a basis for Mk(1), where a

and b range through nonnegative integers such that 4a+6b = k. Moreover, Sk(1) isthe subspace of Mk(1) of elements whose q-expansions have constant coefficient 0.

3.7.2 The Naive Approach

Let k be an even positive integer and p be a prime. Our goal is to compute thecharacteristic polynomial of the Hecke operator Tp,k acting on Sk(1). In practice,when k and p are both reasonably large, e.g., k = 886 and p = 59, then the co-efficients of the characteristic polynomial are huge (the roots of the characteristicpolynomial are O(pk/2−1)). A naive way to compute the characteristic polynomialof Tp,k is to use (3.7.1) to compute the matrix [Tp,k] of Tp,k on the basis of Propo-sition 3.7.1, where E4 and E6 are computed to precision pdimMk(1), and to thencompute the characteristic polynomial of [Tp,k] using, e.g., a modular algorithm(compute the characteristic polynomial modulo many primes, and use the ChineseRemainder Theorem). The difficulty with this approach is that the coefficientsof the q-expansions of Ea4E

b6 to precision pdimMk(1) quickly become enormous,

so both storing them and computing with them is costly, and the components of[Tp,k] are also huge so the characteristic polynomial is difficult to compute. SeeExample 3.2.3 above, where the coefficients of the q-expansions are already large.

3.7.3 The Eigenform Method

We now describe another approach to computing characteristic polynomials, whichgets just the information required. Recall Maeda’s conjecture from Section 3.6.1,which asserts that Sk(1) is spanned by the Gal(Q/Q)-conjugates of a single eigen-form f =

∑bnq

n. For simplicity of exposition below, we assume this conjecture,though the algorithm can probably be modified to deal with the general case. Wewill refer to this eigenform f , which is well-defined up to Gal(Q/Q)-conjugacy, asMaeda’s eigenform.

Lemma 3.7.2. The characteristic polynomial of the pth coefficient bp of Maeda’seigenform f , in the field Q(b1, b2, . . .), is equal to the characteristic polynomial ofTp,k acting on Sk(1).

Proof. The map T⊗Q→ Q(b1, b2, . . .) that sends Tn → bn is an isomorphism ofQ-algebras.

Victor Miller shows in his thesis that Sk(1) has a unique basis f1, . . . , fd ∈ Z[[q]]with ai(fj) = δij , i.e., the first d × d block of coefficients is the identity matrix.Again, in the general case, the requirement that there is such a basis can be avoided,but for simplicity of exposition we assume there is such a basis. We refer to thebasis f1, . . . , fd as Miller’s basis.

Algorithm 3.7.3. We assume in the algorithm that the characteristic polynomialof T2 has no multiple roots (this is easy to check, and if false, you’ve found oninteresting counterexample to the conjecture that the characteristic polynomial ofT2 has Galois group the full symmetric group).

1. Using Proposition 3.7.1 and Gauss elimination, we compute Miller’s basisf1, . . . , fd to precision O(q2d+1), where d = dimSk(1). This is exactly theprecision needed to compute the matrix of T2.


2. Using (3.7.1), we compute the matrix [T2] of T2 with respect to Miller’s basisf1, . . . , fd.

3. Using Algorithm 3.7.5 below we write down an eigenvector e = (e1, . . . , ed) ∈Kd for [T2]. In practice, the components of T2 are not very large, so thenumbers involved in computing e are also not very large.

4. Since e1f1 + · · ·+edfd is an eigenvector for T2, our assumption that the char-acteristic polynomial of T2 is square free (and the fact that T is commutative)implies that e1f1 + · · ·+ edfd is also an eigenvector for Tp. Normalizing, wesee that up to Galois conjugacy,

bp =

d∑

i=1

eie1· ap(fi),

where the bp are the coefficients of Maeda’s eigenform f . For example, sincethe fi are Miller’s basis, if p ≤ d then

bp =epe1

if p ≤ d,

since ap(fi) = 0 for all i 6= p and ap(fp) = 1. Once we have computed bp, wecan compute the characteristic polynomial of Tp, because it is the minimalpolynomial of bp. We spend the rest of this section discussing how to makethis step practical.

Computing bp directly in step 4 is extremely costly because the divisions ei/e1lead to massive coefficient explosion, and the same remark applies to computingthe minimal polynomial of bp. Instead we compute the reductions bp modulo `and the characteristic polynomial of bp modulo ` for many primes `, then recoveronly the characteristic polynomial of bp using the Chinese Remainder Theorem.Deligne’s bound on the magnitude of Fourier coefficients tells us how many primeswe need to work modulo (we leave this analysis to the reader).

More precisely, the reduction modulo ` steps are as follows. The field K can beviewed as Q[x]/(f(x)) where f(x) ∈ Z[x] is the characteristic polynomial of T2.We work only modulo primes such that

1. f(x) has no repeated roots modulo `,

2. ` does not divide any denominator involved in our representation of e, and

3. the image of e1 in F`[x]/(f(x)) is invertible.

For each such prime, we compute the image bp of bp in the reduced Artin ringF`[x]/(f(x)). Then the characteristic polynomial of Tp modulo ` equals the char-acteristic polynomial of bp. This modular arithmetic is fast and requires negligiblestorage. Most of the time is spent doing the Chinese Remainder Theorem com-putations, which we do each time we do a few computations of the characteristicpolynomial of Tp modulo `.

Remark 3.7.4. If k is really large, so that steps 1 and 2 of the algorithm take toolong or require too much memory, steps 1 and 2 can be performed modulo theprime `. Since the characteristic polynomial of Tp,k modulo ` does not depend onany choices, we will still be able to recover the original characteristic polynomial.


3.7.4 How to Write Down an Eigenvector over an Extension Field

The following algorithm, which was suggested to the author by H. Lenstra, pro-duces an eigenvector defined over an extension of the base field.

Algorithm 3.7.5. Let A be an n×n matrix over an arbitrary field k and supposethat the characteristic polynomial f(x) = xn + · · · + a1x+ a0 of A is irreducible.Let α be a root of f(x) in an algebraic closure k of k. Factor f(x) over k(α) asf(x) = (x − α)g(x). Then for any element v ∈ kn the vector g(A)v is either 0 orit is an eigenvector of A with eigenvalue α. The vector g(A)v can be computed byfinding Av, A(Av), A(A(Av)), and then using that

g(x) = xn−1 + cn−2xn−2 + · · ·+ c1x+ c0,

where the coefficients ci are determined by the recurrence

c0 = −a0

α, ci =

ci−1 − aiα

.

We prove below that g(A)v 6= 0 for all vectors v not in a proper subspace ofkn. Thus with high probability, a “randomly chosen” v will have the property thatg(A)v 6= 0. Alternatively, if v1, . . . vn form a basis for kn, then g(A)vi must benonzero for some i.

Proof. By the Cayley-Hamilton theorem [Lan93, XIV.3] we have that f(A) = 0.Consequently, for any v ∈ kn, we have (A − α)g(A)v = 0 so that Ag(A)v =αv. Since f is irreducible it is the polynomial of least degree satisfied by A andso g(A) 6= 0. Therefore g(A)v 6= 0 for all v not in the proper closed subspaceker(g(A)).

3.7.5 Simple Example: Weight 36, p = 3

We compute the characteristic polynomial of T3 acting on S36(1) using the algo-rithm described above. A basis for M36(1) to precision 6 = 2dim(S36(1)) is

E94 = 1 + 2160q + 2093040q2 + 1198601280q3 + 449674832880q4

+ 115759487504160q5 + 20820305837344320q6 +O(q7)

E64E

26 = 1 + 432q − 353808q2 − 257501376q3 − 19281363984q4

+ 28393576094880q5 + 11565037898063424q6 +O(q7)

E34E

46 = 1− 1296q + 185328q2 + 292977216q3 − 52881093648q4

− 31765004621280q5 + 1611326503499328q6 +O(q7)

E66 = 1− 3024q + 3710448q2 − 2309743296q3 + 720379829232q4

− 77533149038688q5 − 8759475843314112q6 +O(q7)

The reduced row-echelon form (Miller) basis is:

f0 = 1 + 6218175600q4 + 15281788354560q5 + 9026867482214400q6 +O(q7)

f1 = q + 57093088q4 + 37927345230q5 + 5681332472832q6 +O(q7)

f2 = q2 + 194184q4 + 7442432q5 − 197264484q6 +O(q7)

f3 = q3 − 72q4 + 2484q5 − 54528q6 +O(q7)


The matrix of T2 with respect to the basis f1, f2, f3 is

[T2] =

0 34416831456 56813324728321 194184 −1972644840 −72 −54528

This matrix has (irreducible) characteristic polynomial

g = x3 − 139656x2 − 59208339456x− 1467625047588864.

If a is a root of this polynomial, then one finds that

e = (2a+ 108984, 2a2 + 108984a, a2 − 394723152a+ 11328248114208)

is an eigenvector with eigenvalue a. The characteristic polynomial of T3 is thenthe characteristic polynomial of e3/e1, which we can compute modulo ` for anyprime ` such that g ∈ F`[x] is square free. For example, when ` = 11,

e3e1

=a2 + a+ 3

2a2 + 7= 9a2 + 2a+ 3,

which has characteristic polynomial

3 + 10x2 + 8x+ 2.

If we repeat this process for enough primes ` and use the Chinese remaindertheorem, we find that the characteristic polynomial of T3 acting on S36(1) is

x3 + 104875308x2 − 144593891972573904x− 21175292105104984004394432.

4Analytic theory of modular curves

4.1 The Modular group

This section very closely follows Sections 1.1–1.2 of [Ser73]. We introduce themodular group G = PSL2(Z), describe a fundamental domain for the action of Gon the upper half plane, and use it to prove that G is generated by

S =

(0 − 11 0

)and T =

(1 10 1

).

4.1.1 The Upper half plane

Leth = z ∈ C : Im(z) > 0

be the open complex upper half plane. The group

SL2(R) =

(a bc d

): a, b, c, d ∈ R and ad− bc = 1

acts by linear fractional transformations (z 7→ (az + b)/(cz + d)) on C ∪ ∞. Bythe following lemma, SL2(R) also acts on h:

i

FIGURE 4.1.1. The upper half plane h

30 4. Analytic theory of modular curves

Lemma 4.1.1. Suppose g ∈ SL2(R) and z ∈ h. Then

Im(gz) =Im(z)

|cz + d|2 .

Proof. Apply the identity Im(z) = 12i (z − z) to both sides of the asserted equality

and simplify.

The only element of SL2(R) that acts trivially on h is −1, so

G = PSL2(R) = SL2(R)/〈−1〉

acts faithfully on h. Let S and T be as above and note that S and T induce thelinear fractional transformations z 7→ −1/z and z 7→ z + 1, respectively. We provebelow that S and T generate G.

4.1.2 Fundamental domain for the modular group

4.1.3 Conjugating an element of the upper half plane into the

fundamental domain

4.1.4 Writing an element in terms of generators

4.1.5 Generators for the modular group

4.2 Congruence subgroups

4.2.1 Definition

4.2.2 Fundamental domains for congruence subgroups

4.2.3 Coset representatives

4.2.4 Generators for congruence subgroups

Simple method

Sophisticated method

4.3 Modular curves

4.3.1 The upper half plane is a disk

4.3.2 The upper half plane union the cusps

4.3.3 The Poincare metric

4.3.4 Fuchsian groups and Riemann surfaces

Definition of Fuchsian group. Quotient of upper half plane.

4.3.5 Riemann surfaces attached to congruence subgroups

X0(N)

4.4 Points on modular curves parameterize elliptic curves with extra structure 31

4.4 Points on modular curves parameterize ellipticcurves with extra structure

The classical theory of the Weierstass ℘-function sets up a bijection betweenisomorphism classes of elliptic curves over C and isomorphism classes of one-dimensional complex tori C/Λ. Here Λ is a lattice in C, i.e., a free abelian groupZω1 + Zω2 of rank 2 such that Rω1 + Rω2 = C.

Any homomorphism ϕ of complex tori C/Λ1 → C/Λ2 is determined by a C-linear map T : C→ C that sends Λ1 into Λ2.

Lemma 4.4.1. Suppose ϕ : C/Λ1 → C/Λ2 is nonzero. Then the kernel of ϕ isisomorphic to Λ2/T (Λ1).

Lemma 4.4.2. Two complex tori C/Λ1 and C/Λ2 are isomorphic if and only ifthere is a complex number α such that αΛ1 = Λ2.

Proof. Any C-linear map C→ C is multiplication by a scalar α ∈ C.

Suppose Λ = Zω1 +Zω2 is a lattice in C, and let τ = ω1/ω2. Then Λτ = Zτ +Zdefines an elliptic curve that is isomorphic to the elliptic curve determined by Λ.By replacing ω1 by −ω1, if necessary, we may assume that τ ∈ h. Thus everyelliptic curve is of the form Eτ = C/Λτ for some τ ∈ h and each τ ∈ h determinesan elliptic curve.

Proposition 4.4.3. Suppose τ, τ ′ ∈ h. Then Eτ ∼= Eτ ′ if and only if there existsg ∈ SL2(Z) such that τ = g(τ ′). Thus the set of isomorphism classes of ellipticcurves over C is in natural bijection with the orbit space SL2(Z)\h.

Proof. Suppose Eτ ∼= Eτ ′ . Then there exists α ∈ C such that αΛτ = Λτ ′ , soατ = aτ ′ + b and α1 = cτ ′ + d for some a, b, c, d ∈ Z. The matrix g =

(a bc d

)has

determinant ±1 since aτ ′ + b and cτ ′ +d form a basis for Zτ +Z; this determinantis positive because g(τ ′) = τ and τ, τ ′ ∈ h. Thus det(g) = 1, so g ∈ SL2(Z).

Conversely, suppose τ, τ ′ ∈ h and g =(a bc d

)∈ SL2(Z) is such that

τ = g(τ ′) =aτ ′ + b

cτ ′ + d.

Let α = cτ ′ + d, so ατ = aτ ′ + b. Since det(g) = 1, the scalar α defines anisomorphism from Λτ to Λτ ′ , so Eτ ∼= E′

τ , as claimed.

Let E = C/Λ be an elliptic curve over C and N a positive integer. UsingLemma 4.4.1, we see that

E[N ] := x ∈ E : Nx = 0 ∼=(

1

NΛ

)/Λ ∼= (Z/NZ)2.

If Λ = Λτ = Zτ + Z, this means that τ/N and 1/N are a basis for E[N ].Suppose τ ∈ h and recall that Eτ = C/Λτ = C/(Zτ + Z). To τ , we associate

three “level N structures”. First, let Cτ be the subgroup of Eτ generated by 1/N .Second, let Pτ be the point of order N in Eτ defined by 1/N ∈ Λτ . Third, let Qτbe the point of order N in Eτ defined by τ/N , and consider the basis (Pτ , Qτ ) forE[N ].


In order to describe the third level structure, we introduce the Weil pairing

e : E[N ]× E[N ]→ Z/NZ

as follows. If E = C/(Zω1 + Zω2) with ω1/ω2 ∈ h, and P = aω1/N + bω2/N ,Q = cω1/N + dω2/N , then

e(P,Q) = ad− bc ∈ Z/NZ.

Notice that e(Pτ , Qτ ) = −1 ∈ Z/NZ. Also if C/Λ ∼= C/Λ′ via multiplication by α,and P,Q ∈ (C/Λ)[N ], then e(α(P ), α(Q)) = e(P,Q).

Theorem 4.4.4. Let N be a positive integer.

1. The non-cuspidal points on X0(N) correspond to isomorphism classes ofpairs (E,C) where C is a cyclic subgroup of E of order N . (Two pairs(E,C), (E′, C ′) are isomorphic if there is an isomorphism ϕ : E → E ′

such that ϕ(C) = C ′.)

2. The non-cuspidal points on X1(N) correspond to pairs (E,P ) where P is apoint on E of exact order N . (Two pairs (E,P ) and (E ′, P ′) isomorphic ifthere is an isomorphism ϕ : E → E′ such that ϕ(P ) = P ′.)

3. The non-cuspidal points on X(N) correspond to triples (E,P,Q) where P,Qare a basis for E[N ] such that e(P,Q) = −1 ∈ Z/NZ. (Triples (E,P,Q) and(E,P ′, Q′) are isomorphic if there is an isomorphism ϕ : E → E ′ such thatϕ(P ) = P ′ and ϕ(Q) = Q′.)

This theorem follows from Propositions 4.4.5 and 4.4.7 below.

Proposition 4.4.5. Let E be an elliptic curve over C. If C is a cyclic subgroup ofE of order N , then there exists τ ∈ h such that (E,C) is isomorphic to (Eτ , Cτ ).If P is a point on E of order N , then there exists τ ∈ C such that (E,P ) isisomorphic to (Eτ , Pτ ). If P,Q is a basis for E[N ] and e(P,Q) = −1 ∈ Z/NZ,then there exists τ ∈ C such that (E,P,Q) is isomorphic to (Eτ , Pτ , Qτ ).

Proof. Write E = C/Λ with Λ = Zω1 + Zω2 and ω1/ω2 ∈ h.Suppose P = aω1/N + bω2/N is a point of order N . Then gcd(a, b,N) = 1,

otherwise P would have order strictly less than N , a contradiction. Thus we canmodify a and b by adding multiples of N to them (this follows from the fact thatSL2(Z)→ SL2(Z/NZ) is surjective), so that P = aω1/N + bω2/N and gcd(a, b) =1. There exists c, d ∈ Z such that ad−bc = 1, so ω′

1 = aω1+bω2 and ω′2 = cω1+dω2

form a basis for Λ, and C is generated by P = ω′1/N . If necessary, replace ω′

2 by−ω′

2 so that τ = ω′2/ω

′1 ∈ h. Then (E,P ) is isomorphic to (Eτ , Pτ ). Also, if C is

the subgroup generated by P , then (E,C) is isomorphic to (Eτ , Cτ ).Suppose P = aω1/N + bω2/N and Q = cω1/N + dω2/N are a basis for E[N ]

with e(P,Q) = −1. Then the matrix(a b−c −d

)has determinant 1 modulo N , so

because the map SL2(Z) → SL2(Z/NZ) is surjective, we can replace a, b, c, d byintegers which are equivalent to them modulo N (so P and Q are unchanged) andso that ad − bc = −1. Thus ω′

1 = aω1 + bω2 and ω′2 = cω1 + dω2 form a basis for

Λ. Let

τ = ω′2/ω

′1 =

cω1

ω2+ d

aω1

ω2+ b

.

4.5 The Genus of X(N) 33

Then τ ∈ h since ω1/ω2 ∈ h and(c da b

)has determinant +1. Finally, division by

ω′1 defines an isomorphism E → Eτ that sends P to 1/N and Q to τ/N .

Remark 4.4.6. Part 3 of Theorem 2.4 in Chapter 11 of Husemoller’s book onelliptic curves is wrong, since he neglects the Weil pairing condition. Also thefirst paragraph of his proof of the theorem is incomplete.

The following proposition completes the proof of Theorem 4.4.4.

Proposition 4.4.7. Suppose τ, τ ′ ∈ h. Then (Eτ , Cτ ) is isomorphic (Eτ ′ , Cτ ′) ifand only if there exists g ∈ Γ0(N) such that g(τ) = τ ′. Also, (Eτ , Pτ ) is isomorphic(Eτ ′ , Pτ ′) if and only if there exists g ∈ Γ1(N) such that g(τ) = τ ′. Finally,(Eτ , Pτ , Qτ ) is isomorphic (Eτ ′ , Pτ ′ , Qτ ′) if and only if there exists g ∈ Γ(N) suchthat g(τ) = τ ′.

Proof. We prove only the first assertion, since the others are proved in a similarway. Suppose (Eτ , Cτ ) is isomorphic to (E′

τ , C′τ ). Then there is λ ∈ C such that

λΛτ = Λτ ′ . Thus λτ = aτ ′ + b and λ1 = cτ ′ + d with g =(a bc d

)∈ SL2(Z) (as

we saw in the proof of Proposition 4.4.3). Dividing the second equation by N weget λ 1

N = cN τ

′ + dN , which lies in Λτ ′ = Zτ ′ + 1

NZ, by hypothesis. Thus c ≡ 0(mod N), so g ∈ Γ0(N), as claimed. For the converse, note that if N | c, thencN τ

′ + dN ∈ Λτ ′ .

4.5 The Genus of X(N)

Let N be a positive integer. The aim of this section is to establish some factsabout modular curves associated to congruence subgroups and compute the genusof X(N). Similar methods can be used to compute the genus of X0(N) and X1(N)(for X0(N) see [Shi94, §1.6] and for X1(N) see [DI95, §9.1]).

The groups Γ0(1), Γ1(1), and Γ(1) are all equal to SL2(Z), so X0(1) = X1(1) =X(1) = P1. Since P1 has genus 0, we know the genus for each of these threecases. For general N we obtain the genus by determining the ramification of thecorresponding cover of P1 and applying the Hurwitz formula, which we assumethe reader is familiar with, but which we now recall.

Suppose f : X → Y is a surjective morphism of Riemann surfaces of degree d.For each point x ∈ X, let ex be the ramification exponent at x, so ex = 1 preciselywhen f is unramified at x, which is the case for all but finitely many x. (There is apoint over y ∈ Y that is ramified if and only if the cardinality of f−1(y) is less thanthe degree of f .) Let g(X) and g(Y ) denote the genera of X and Y , respectively.

Theorem 4.5.1 (Hurwitz Formula). Let f : X → Y be as above. Then

2g(X)− 2 = d(2g(Y )− 2) +∑

x∈X(ex − 1).

If X → Y is Galois, so the ex in the fiber over each fixed y ∈ Y are all equal, thenthis formula becomes

2g(X)− 2 = d

2g(Y )− 2 +

∑

y∈Y

(1− 1

ey

) .


Let X be one of the modular curves X0(N), X1(N), or X(N) corresponding to acongruence subgroup Γ, and let Y = X(1) = P1. There is a natural map f : X → Ygot by sending the equivalence class of τ modulo the congruence subgroup Γ to theequivalence class of τ modulo SL2(Z). This is “the” map X → P1 that we meaneverywhere below.

Because PSL2(Z) acts faithfully on h, the degree of f is the index in PSL2(Z)of the image of Γ in PSL2(Z) (see Exercise X). Using that the map SL2(Z) →SL2(Z/NZ) is surjective, we can compute these indices (Exercise X), and obtainthe following lemma:

Proposition 4.5.2. Suppose N > 2. The degree of the map X0(N) → P1 isN∏p|N (1 + 1/p). The degree of the map X1(N) → P1 is 1

2N2∏p|N (1 − 1/p2).

The degree of the map from X(N) → P1 is 12N

3∏p|N (1 − 1/p2). If N = 2, then

the degrees are 3, 3, and 6, respectively.

Proof. This follows from the discussion above, Exercise X about indices of congru-ence subgroups in SL2(Z), and the observation that for N > 2 the groups Γ(N)and Γ1(N) do not contain −1 and the group Γ0(N) does.

Proposition 4.5.3. Let X be X0(N), X1(N) or X(N). Then the map X → P1

is ramified at most over ∞ and the two points corresponding to elliptic curves withextra automorphisms (i.e., the two elliptic curves with j-invariants 0 and 1728).

Proof. Since we have a tower X(N)→ X1(N)→ X0(N)→ P1, it suffices to provethe assertion for X = X(N). Since we do not claim that there is no ramificationover ∞, we may restrict to Y (N). By Theorem 4.4.4, the points on Y (N) cor-respond to isomorphism classes of triples (E,P,Q), where E is an elliptic curveover C and P,Q are a basis for E[N ]. The map from Y (N) to P1 sends the iso-morphism class of (E,P,Q) to the isomorphism class of E. The equivalence classof (E,P,Q) also contains (E,−P,−Q), since −1 : E → E is an isomorphism. Theonly way the fiber over E can have cardinality smaller than the degree is if thereis an extra equivalence (E,P,Q) → (E,ϕ(P ), ϕ(Q)) with ϕ an automorphism ofE not equal to ±1. The theory of CM elliptic curves shows that there are only twoisomorphism classes of elliptic curves E with automorphisms other than ±1, andthese are the ones with j-invariant 0 and 1728. This proves the proposition.

Theorem 4.5.4. For N > 2, the genus of X(N) is

g(X(N)) = 1 +N2(N − 6)

24

∏

p|N

(1− 1

p2

).

For N = 1, 2, the genus is 0.

Thus if gN = g(X(N)), then g1 = g2 = g3 = g4 = g5 = 0, g6 = 1, g7 = 3, g8 = 5,g9 = 10, g389 = 2414816, and g2003 = 333832500.

Proof. Since X(N) is a Galois covering of X(1) = P1, the ramification indices exare all the same for x over a fixed point y ∈ P1; we denote this common indexby ey. The fiber over the curve with j-invariant 0 has size one-third of the degree,since the automorphism group of the elliptic curve with j-invariant 0 has order 6,so the group of automorphisms modulo ±1 has order three, hence e0 = 3. Similarly,the fiber over the curve with j-invariant 1728 has size half the degree, since the

4.5 The Genus of X(N) 35

automorphism group of the elliptic curve with j-invariant 1728 is cyclic of order 4,so e1728 = 2.

To compute the ramification degree e∞ we use the orbit stabilizer theorem.The fiber of X(N) → X(1) over ∞ is exactly the set of Γ(N) equivalence classesof cusps, which is Γ(N)∞,Γ(N)g2∞, . . . ,Γ(N)gr∞, where g1 = 1, g2, . . . , gr arecoset representatives for Γ(N) in SL2(Z). By the orbit-stabilizer theorem, thenumber of cusps equals #(Γ(1)/Γ(N))/#S, where S is the stabilizer of Γ(N)∞in Γ(1)/Γ(N) ∼= SL2(Z/NZ). Thus S is the subgroup ± ( 1 n

0 1 ) : 0 ≤ n < N − 1,which has order 2N . Since the degree of X(N) → X(1) equals #(Γ(1)/Γ(N))/2,the number of cusps is the degree divided by N . Thus e∞ = N .

The Hurwitz formula for X(N)→ X(1) with e0 = 3, e1728 = 2, and e∞ = N , is

2g(X(N))− 2 = d

(0− 2 +

(1− 1

3+ 1− 1

2+ 1− 1

N

)),

where d is the degree of X(N)→ X(1). Solving for g(X(N)) we obtain

2g(X)− 2 = d

(1− 5

6− 1

N

)= d

(N − 6

6N

),

so

g(X) = 1 +d

2

(N − 6

6N

)=

d

12N(N − 6) + 1.

Substituting the formula for d from Proposition 4.5.2 yields the claimed formula.

5Modular Symbols

This chapter is about how to explicitly compute the homology of modular curvesusing modular symbols.

We assume the reader is familiar with basic notions of algebraic topology, in-cluding homology groups of surfaces and triangulation. We also assume that thereader has read XXX about the fundamental domain for the action of PSL2(Z) onthe upper half plane, and XXX about the construction of modular curves.

Some standard references for modular symbols are [Man72] [Lan95, IV], [Cre97],and [Mer94]. Sections 5.1–5.2 below very closely follow Section 1 of Manin’s paper[Man72].

For the rest of this chapter, let Γ = PSL2(Z) and let G be a subgroup of Γof finite index. Note that we do not require G to be a congruence subgroup. Thequotient X(G) = G\h∗ of h∗ = h ∪ P1(Q) by G has an induced structure ofcompact Riemann surface. Let π : h∗ → X(G) denote the natural projection. Thematrices

s =

(0 − 11 0

)and t =

(1 − 11 0

)

together generate Γ; they have orders 2 and 3, respectively.

5.1 Modular symbols

Let H0(X(G),Ω1) denote the complex vector space of holomorphic 1-forms onX(G). Integration of differentials along homology classes defines a perfect pairing

H1(X(G),R)×H0(X(G),Ω1)→ C,

hence an isomorphism

H1(X(G),R) ∼= HomC(H0(X(G),Ω1),C).

38 5. Modular Symbols

For more details, see [Lan95, §IV.1].Given two elements α, β ∈ h∗, integration from α to β induces a well-defined

element of HomC(H0(X(G),Ω1),C), hence an element

α, β ∈ H1(X(G),R).

Definition 5.1.1 (Modular symbol). The homology class α, β ∈ H1(X(G),R)associated to α, β ∈ h∗ is called the modular symbol attached to α and β.

Proposition 5.1.2. The symbols α, β have the following properties:

1. α, α = 0, α, β = −β, α, and α, β+ β, γ+ γ, α = 0.

2. g(α), g(β) = α, β for all g ∈ G

3. If X(G) has nonzero genus, then α, β ∈ H1(X(G),Z) if and only if G(α) =G(β) (i.e., the cusps α and β are equivalent).

Remark 5.1.3. We only have α, β = β, α if α, β = 0, so the modular symbolsnotation, which suggests “unordered pairs”, is actively misleading.

Proposition 5.1.4. For any α ∈ h∗, the map G → H1(X(G),Z) that sends g toα, gα is a surjective group homomorphism that does not depend on the choiceof α.

Proof. If g, h ∈ G and α ∈ h∗, then

α, gh(α) = α, gα+ gα, ghα = α, gα+ α, hα,

so the map is a group homomorphism. To see that the map does not depend onthe choice of α, suppose β ∈ h∗. By Proposition 5.1.2, we have α, β = gα, gβ.Thus

α, gα+ gα, β = gα, β+ β, gβ,so cancelling gα, β from both sides proves the claim.

The fact that the map is surjective follows from general facts from algebraictopology. Let h0 be the complement of Γi ∪ Γρ in h, fix α ∈ h0, and let X(G)0 =π(h0). The map h0 → X(G)0 is an unramified covering of (noncompact) Riemannsurfaces with automorphism group G. Thus α determines a group homomorphismπ1(X(G)0, π(α)) → G. When composed with the morphism G → H1(X(G),Z)above, the composition

π1(X(G)0, π(α))→ G→ H1(X(G),Z)

is the canonical map from the fundamental group of X(G)0 to the homology ofthe corresponding compact surface, which is surjective. This forces the map G→H1(X(G),Z) to be surjective, which proves the claim.

5.2 Manin symbols

We continue to assume that G is a finite-index subgroup of Γ = PSL2(Z), so theset G\Γ = Gg1, . . . Ggd of right cosets of G in Γ is finite.

5.2 Manin symbols 39

5.2.1 Using continued fractions to obtain surjectivity

Let R = G\Γ be the set of right cosets of G in Γ. Define

[ ] : R→ H1(X(G),R)

by [r] = r0, r∞, where r0 means the image of 0 under any element of the cosetr (it doesn’t matter which). For g ∈ Γ, we also write [g] = [gG].

Proposition 5.2.1. Any element of H1(X(G),Z) is a sum of elements of the form[r], and the representation

∑nrαr, βr of h ∈ H1(X(G),Z) can be chosen so that∑

nr(π(βr)− π(αr)) = 0 ∈ Div(X(G)).

Proof. By Proposition 5.1.4, every element h of H1(X(G),Z) is of the form 0, g(0)for some g ∈ G. If g(0) = ∞, then h = [G] and π(∞) = π(0), so we may assumeg(0) = a/b 6=∞, with a/b in lowest terms and b > 0. Also assume a > 0, since thecase a < 0 is treated in the same way. Let

0 =p−2

q−2=

0

1,p−1

q−1=

1

0,p0

1=p0

q0,p1

q1,p2

q2, . . . ,

pnqn

=a

b

denote the continued fraction convergents of the rational number a/b. Then

pjqj−1 − pj−1qj = (−1)j−1 for − 1 ≤ j ≤ n.

If we let gj =

((−1)j−1pj pj−1

(−1)j−1qj qj−1

), then gj ∈ SL2(Z) and

0,a

b

=

n∑

j=−1

pj−1

qj−1,pjqj

=

n∑

j=−1

gj0, gj∞)

=

r∑

j=−1

[gj ].

For the assertion about the divisor sum equaling zero, notice that the endpointsof the successive modular symbols cancel out, leaving the difference of 0 and g(0)in the divisor group, which is 0.

Lemma 5.2.2. If x =∑tj=1 njαj , βj is a Z-linear combination of modular

symbols for G and∑nj(π(βj)− π(αj)) = 0 ∈ Div(X(G)), then x ∈ H1(X(G),Z).

Proof. We may assume that each nj is ±1 by allowing duplication. We may furtherassume that each nj = 1 by using that α, β = −β, α. Next reorder the sum soπ(βj) = π(αj+1) by using that the divisor is 0, so every βj must be equivalent tosome αj′ , etc. The lemma should now be clear.


i 1+i

10

oo

E'

t2E' tE'

(1+i)/2

rho

FIGURE 5.2.1.

5.2.2 Triangulating X(G) to obtain injectivity

Let C be the abelian group generated by symbols (r) for r ∈ G\Γ, subject to therelations

(r) + (rs) = 0, and (r) = 0 if r = rs.

For (r) ∈ C, define the boundary of (r) to be the difference π(r∞) − π(r0) ∈Div(X(G)). Since s swaps 0 and ∞, the boundary map is a well-defined map onC. Let Z be its kernel.

Let B be the subgroup of C generated by symbols (r), for all r ∈ G\Γ thatsatisfy r = rt, and by (r) + (rt) + (rt2) for all other r. If r = rt, then rt(0) = r(0),so r(∞) = r(0), so (r) ∈ Z. Also, using (5.2.1) below, we see that for any r, theelement (r) + (rt) + (rt2) lies in Z.

The map G\Γ → H1(X(G),R) that sends (r) to [r] induces a homomorphismC → H1(X(G),R), so by Proposition 5.2.1 we obtain a surjective homomorphism

ψ : Z/B → H1(X(G),Z).

Theorem 5.2.3 (Manin). The map ψ : Z/B → H1(X(G),Z) is an isomorphism.

Proof. We only have to prove that ψ is injective. Our proof follows the proofof [Man72, Thm. 1.9] very closely. We compute the homology H1(X(G),Z) bytriangulating X(G) to obtain a simplicial complex L with homology Z1/B1, thenembed Z/B in the homology Z1/B1 of X(G). Most of our work is spent describingthe triangulation L.

Let E denote the interior of the triangle with vertices 0, 1, and∞, as illustratedin Figure 5.2.1. Let E′ denote the union of the interior of the region bounded bythe path from i to ρ = eπi/3 to 1+ i to ∞ with the indicated path from i to ρ, notincluding the vertex i.

When reading the proof below, it will be helpful to look at the following ta-ble, which illustrates what s =

(0 −11 0

), t =

(1 −11 0

), and t2 do to the vertices in


Figure 5.2.1:

1 0 1 ∞ i 1 + i (1 + i)/2 ρ

s ∞ −1 0 i (−1 + i)/2 −1 + i −ρt ∞ 0 1 1 + i (1 + i)/2 i ρ

t2 1 ∞ 0 (1 + i)/2 i 1 + i ρ

(5.2.1)

Note that each of E′, tE′, and t2E′ is a fundamental domain for Γ, in the sensethat every element of the upper half plane is conjugate to exactly one element inthe closure of E′ (except for identifications along the boundaries). For example,E′ is obtained from the standard fundamental domain for Γ, which has verticesρ2, ρ, and ∞, by chopping it in half along the imaginary axis, and translating thepiece on the left side horizontally by 1.

If (0,∞) is the path from 0 to ∞, then t(0,∞) = (∞, 1) and t2(0,∞) = (1, 0).Also, s(0,∞) = (∞, 0). Thus each half side of E is Γ-conjugate to the side fromi to ∞. Also, each 1-simplex in Figure 5.2.1, i.e., the sides that connected twoadjacent labeled vertices such as i and ρ, maps homeomorphically into X(Γ). Thisis clear for the half sides, since they are conjugate to a path in the interior of thestandard fundamental domain for Γ, and for the medians (lines from midpoints toρ) since the path from i to ρ is on an edge of the standard fundamental domainwith no self identifications.

We now describe our triangulation L of X(G):

0-cells The 0 cells are the cusps π(P1(Q)) and i-elliptic points π(Γi). Note thatthese are the images under π of the vertices and midpoints of sides of thetriangles gE, for all g ∈ Γ.

1-cells The 1 cells are the images of the half-sides of the triangles gE, for g ∈ Γ,oriented from the edge to the midpoint (i.e., from the cusp to the i-ellipticpoint). For example, if r = Gg is a right coset, then

e1(r) = π(g(∞), g(i)) ∈ X(G)

is a 1 cell in L. Since, as we observed above, every half side is Γ-conjugateto e1(G), it follows that every 1-cell is of the form e(r) for some right cosetr ∈ G\Γ.

Next observe that if r 6= r′ then

e1(r) = e1(r′) implies r′ = rs. (5.2.2)

Indeed, if π(g(∞), g(i)) = π(g′(∞), g′(i)), then ri = r′i (note that the end-points of a path are part of the definition of the path). Thus there existsh, h′ ∈ G such that hg(i) = h′g′(i). Since the only nontrivial element of Γthat stabilizes i is s, this implies that (hg)−1h′g′ = s. Thus h′g′ = hgs, soGg′ = Ggs, so r′ = rs.

2-cells There are two types of 2-cells, those with 2 sides and those with 3.

2-sided: The 2-sided 2-cells e2(r) are indexed by the cosets r = Gg suchthat rt = r. Note that for such an r, we have π(rE ′) = π(rtE′) = π(rt2E′).The 2-cell e2(r) is π(gE′). The image g(ρ, i) of the half median maps to a


line from the center of e2(r) to the edge π(g(i)) = π(g(1 + i)). Orient e2(r)in a way compatible with the e1. Since Ggt = Gg,

π(g(1 + i), g(∞)) = π(gt2(1 + i), gt2(∞)) = π(g(i), g(0)) = π(gs(i), gs(∞)),

so

e1(r)−e1(rs) = π(g(∞), g(i))+π(gs(i), gs(∞)) = π(g(∞), g(i))+π(g(1+i), g(∞)).

Thus∂e2(r) = e1(r)− e1(rs).

Finally, note that if r′ 6= r also satisfies r′t = r′, then e2(r) 6= e2(r′) (to see

this use that E′ is a fundamental domain for Γ).

3-sided: The 3-sided 2-cells e2(r) are indexed by the cosets r = Gg suchthat rt 6= r. Note that for such an r, the three triangles rE ′, rtE′, and rt2E′

are distinct (since they are nontrivial translates of a fundamental domain).Orient e2(r) in a way compatible with the e1 (so edges go from cusps tomidpoints). Then

∂e2(r) =

2∑

n=0

(e1(rtn)− e1(rtns)) .

We have now defined a complex L that is a triangulation of X(G). Let C1, Z1,and B1 be the group of 1-chains, 1-cycles, and 1-boundaries of the complex L.Thus C1 is the abelian group generated by the paths e1(r), the subgroup Z1 is thekernel of the map that sends e1(r) = π(r(∞), r(0)) to π(r(0)) − π((∞)), and B1

is the subgroup of Z1 generated by boundaries of 2-cycles.Let C,Z,B be as defined before the statement of the Theorem 5.2.3. We have

H1(X(G),Z) ∼= Z1/B1, and would like to prove that Z/B ∼= Z1/B1.Define a map ϕ : C → C1 by (r) 7→ e1(rs) − e1(r). The map ϕ is well defined

because if r = rs, then clearly (r) 7→ 0, and (r) + (rs) maps to e1(rs) − e1(r) +e1(r) − e1(rs) = 0. To see that f is injective, suppose

∑nr(r) 6= 0. Since in C

we have the relations (r) = −(rs) and (r) = 0 if rs = r, we may assume thatnrnrs = 0 for all r. We have

ϕ(∑

nr(r))

=∑

nr(e1(rs)− e1(r)).

If nr 6= 0 then r 6= rs, so (5.2.2) implies that e1(r) 6= e1(rs). If nr 6= 0 and nr′ 6= 0with r′ 6= r, then r 6= rs and r′ 6= r′s, so e1(r), e1(rs), e1(r

′), e1(r′s) are all distinct.We conclude that

∑nr(e1(rs)− e1(r)) 6= 0, which proves that ϕ is injective.

Suppose (r) ∈ C. Then

ϕ(r) +B1 = ψ(r) = r(0), r(∞) ∈ H1(X(G),Z) = C1/B1,

since

ϕ(r) = e1(rs)−e1(r) = π(rs(∞), rs(i))−π(r(∞), r(i)) = π(r(0), r(i))−π(r(∞), r(i))

belongs to the homology class r(0), r(∞). Extending linearly, we have, for anyz ∈ C, that ϕ(z) +B1 = ψ(z).


The generators for B1 are the boundaries of 2-cells e2(r). As we saw above, thesehave the form ϕ(r) for all r such that r = rt, and ϕ(r) + ϕ(rt) + ϕ(rt2) for ther such that rt 6= r. Thus B1 = ϕ(B) ⊂ ϕ(Z), so the map ϕ induces an injectionZ/B → Z1/B1. This completes the proof of the theorem.


5.3 Hecke Operators

In this section we will only consider the modular curve X0(N) associated to thesubgroup Γ0(N) of SL2(Z) of matrices that are upper triangular modulo N . Muchof what we say will also be true, possibly with slight modification, for X1(N), butnot for arbitrary finite-index subgroups.

There is a commutative ring

T = Z[T1, T2, T3, . . .]

of Hecke operators that acts on H1(X0(N),R). We will frequently revisit this ring,which also acts on the Jacobian J0(N) of X0(N), and on modular forms. The ringT is generated by Tp, for p prime, and as a free Z-module T is isomorphic to Zg,where g is the genus of X0(N). We will not prove these facts here (see ).

Supposeα, β ∈ H1(X0(N),R),

is a modular symbol, with α, β ∈ P1(Q). For g ∈ M2(Z), write g(α, β) =g(α), g(β). This is not a well-defined action of M2(Z) on H1(X0(N),R), sinceα′, β′ = α, β ∈ H1(X0(N),R) does not imply that g(α′), g(β′) = g(α), g(β).Example 5.3.1. Using Magma we see that the homology H1(X0(11),R) is gener-ated by −1/7, 0 and −1/5, 0.> M := ModularSymbols(11); // Homology relative to cusps,

// with Q coefficients.

> S := CuspidalSubspace(M); // Homology, with Q coefficients.

> Basis(S);

[ -1/7, 0, -1/5, 0 ]

Also, we have 50,∞ = −1/5, 0.> pi := ProjectionMap(S); // The natural map M --> S.

> M.3;

oo, 0

> pi(M.3);

-1/5*-1/5, 0

Let g = ( 2 00 1 ). Then 5g(0), g(∞) is not equal to g(−1/5), g(0), so g does not

define a well-defined map on H1(X0(11),R).

> x := 5*pi(M!<1,[Cusps()|0,Infinity()]>);

> y := pi(M!<1,[-2/5,0]>);

> x;

-1/5, 0

> y;

-1*-1/7, 0 + -1*-1/5, 0

> x eq y;

false

Definition 5.3.2 (Hecke operators). We define the Hecke operator Tp onH1(X0(N),R) as follows. When p is a prime with p - N , we have

Tp(α, β) =

(p 00 1

)(α, β) +

p−1∑

r=0

(1 r0 p

)(α, β).

5.4 Modular Symbols and Rational Homology 45

When p | N , the formula is the same, except that the first summand, which involves(p 00 1

), is omitted.

Example 5.3.3. We continue with Example 5.3.1. If we apply the Hecke operatorT2 to both 50,∞ and −1/5, 0, the “non-well-definedness” cancels out.

> x := 5*pi(M!<1,[Cusps()|0,Infinity()]> +

M!<1,[Cusps()|0,Infinity()]> + M!<1,[Cusps()|1/2,Infinity()]>);

> x;

-2*-1/5, 0

> y := pi(M!<1,[-2/5,0]>+ M!<1,[-1/10,0]> + M!<1,[2/5,1/2]>);

> y;

-2*-1/5, 0

Examples 5.3.1 shows that it is not clear that the definition of Tp given abovemakes sense. For example, if α, β is replaced by an equivalent modular symbolα′, β′, why does the formula for Tp give the same answer? We will not addressthis question further here, but will revisit it later when we have a more naturaland intrinsic definition of Hecke operators. We only remark that Tp is induced bya “correspondence” from X0(N) to X0(N), so Tp preserve H1(X0(N),Z).

5.4 Modular Symbols and Rational Homology

In this section we sketch a beautiful proof, due to Manin, of a result that is crucialto our understanding of rationality properties of special values of L-functions. Forexample, Mazur and Swinnerton-Dyer write in [MSD74, §6], “The modular symbolis essential for our theory of p-adic Mellin transforms,” right before discussing thisrationality result. Also, as we will see in the next section, this result implies that ifE is an elliptic curve over Q, then L(E, 1)/ΩE ∈ Q, which confirms a consequenceof the Birch and Swinnerton-Dyer conjecture.

Theorem 5.4.1 (Manin). For any α, β ∈ P1(Q), we have

α, β ∈ H1(X0(N),Q).

Proof (sketch). Since α, β = α,∞−β,∞, it suffices to show that α,∞ ∈H1(X0(N),Q) for all α ∈ Q. We content ourselves with proving that 0,∞ ∈H1(X0(N),Z), since the proof for general 0, α is almost the same.

We will use that the eigenvalues of Tp on H1(X0(N),R) have absolute valuebounded by 2

√p, a fact that was proved by Weil (the Riemann hypothesis for

curves over finite fields). Let p - N be a prime. Then

Tp(0,∞) = 0,∞+

p−1∑

r=0

r

p,∞

= (1 + p)0,∞+

p−1∑

r=0

r

p, 0

,

so

(1 + p− Tp)(0,∞) =

p−1∑

r=0

0,r

p

.

Since p - N , the cusps 0 and r/p are equivalent (use the Euclidean algorithmto find a matrix in SL2(Z) of the form ( r ∗

p ∗ )), so the modular symbols 0, r/p,


for r = 0, 1, . . . , p − 1 all lie in H1(X0(N),Z). Since the eigenvalues of Tp haveabsolute value at most 2

√p, the linear transformation 1 + p− Tp of H1(X0(N),Z)

is invertible. It follows that some integer multiple of 0,∞ lies in H1(X0(N),Z),as claimed.

There are general theorems about the denominator of α, β in some cases.Example 5.3.1 above demonstrated the following theorem in the case N = 11.

Theorem 5.4.2 (Ogg [Ogg71]). Let N be a prime. Then the image

[0,∞] ∈ H1(X0(N),Q)/H1(X0(N),Z)

has order equal to the numerator of (N − 1)/12.

5.5 Special Values of L-functions

This section is a preview of one of the central arithmetic results we will discuss inmore generality later in this book.

The celebrated modularity theorem of Wiles et al. asserts that there is a cor-respondence between isogeny classes of elliptic curves E of conductor N and nor-malized new modular eigenforms f =

∑anq

n ∈ S2(Γ0(N)) with an ∈ Z. Thiscorrespondence is characterized by the fact that for all primes p - N , we haveap = p+ 1−#E(Fp).

Recall that a modular form for Γ0(N) of weight 2 is a holomorphic functionf : h→ C that is “holomorphic at the cusps” and such that for all

(a bc d

)∈ Γ0(N),

f

(az + b

cz + d

)= (cz + d)2f(z).

Suppose E is an elliptic curve that corresponds to a modular form f . If L(E, s)is the L-function attached to E, then

L(E, s) = L(f, s) =∑ an

ns,

so, by a theorem of Hecke which we will prove [later], L(f, s) is holomorphic onall C. Note that L(f, s) is the Mellin transform of the modular form f :

L(f, s) = (2π)sΓ(s)−1

∫ i∞

0

(−iz)sf(z)dz

z. (5.5.1)

The Birch and Swinnerton-Dyer conjecture concerns the leading coefficient ofthe series expansion of L(E, s) about s = 1. A special case is that if L(E, 1) 6= 0,then

L(E, 1)

ΩE=

∏cp ·#X(E)

#E(Q)2tor.

Here ΩE = |∫E(R)

ω|, where ω is a “Neron” differential 1-form on E, i.e., a gen-

erator for H0(E ,Ω1E/Z), where E is the Neron model of E. (The Neron model of

E is the unique, up to unique isomorphism, smooth group scheme E over Z, withgeneric fiber E, such that for all smooth schemes S over Z, the natural mapHomZ(S, E)→ HomQ(S×Spec(Q), E) is an isomorphism.) In particular, the con-jecture asserts that for any elliptic curve E we have L(E, 1)/ΩE ∈ Q.

5.5 Special Values of L-functions 47

Theorem 5.5.1. Let E be an elliptic curve over Q. Then L(E, 1)/ΩE ∈ Q.

Proof (sketch). By the modularity theorem of Wiles et al., E is modular, so thereis a surjective morphism πE : X0(N) → E, where N is the conductor of E. Thisimplies that there is a newform f that corresponds to (the isogeny class of) E, withL(f, s) = L(E, s). Also assume, without loss of generality, that E is “optimal” inits isogeny class, which means that if X0(N)→ E′ → E is a sequence of morphismwhose composition is πE and E′ is an elliptic curve, then E′ = E.

By Equation 5.5.1, we have

L(E, 1) = 2π

∫ i∞

0

−izf(z)dz/z. (5.5.2)

If q = e2πiz, then dq = 2πiqdz, so 2πif(z)dz = dq/q, and (5.5.2) becomes

L(E, 1) = −∫ i∞

0

f(q)dq.

Recall that ΩE = |∫E(R)

ω|, where ω is a Neron differential on E. The expression

f(q)dq defines a differential on the modular curve X0(N), and there is a rationalnumber c, the Manin constant, such that π∗

Eω = cf(q)dq. More is true: Edixhovenproved (as did Ofer Gabber) that c ∈ Z; also Manin conjectured that c = 1 andEdixhoven proved (unpublished) that if p | c, then p = 2, 3, 5, 7.

A standard fact is that if

L =

∫

γ

ω : γ ∈ H1(E,Z)

is the period lattice of E associated to ω, then E(C) ∼= C/L. Note that ΩE iseither the least positive real element of L or twice this least positive element (ifE(R) has two real components).

The next crucial observation is that by Theorem 5.4.1, there is an integer n suchthat n0,∞ ∈ H1(X0(N),Z). This is relevant because if

L′ =

∫

γ

f(q)dq : γ ∈ H1(X0(N),Z)

⊂ C.

then L = 1cL′ ⊂ L′. This assertion follows from our hypothesis that E is optimal

and standard facts about complex tori and Jacobians, which we will prove later[in this course/book].

One can show that L(E, 1) ∈ R, for example, by writing down an explicit realconvergent series that converges to L(E, 1). This series is used in algorithms tocompute L(E, 1), and the derivation of the series uses properties of modular formsthat we have not yet developed. Another approach is to use complex conjugationto define an involution ∗ on H1(X0(N),R), then observe that 0,∞ is fixed by∗. (The involution ∗ is given on modular symbols by ∗α, β = −α,−β.)

Since L(E, 1) ∈ R, the integral

∫

n0,∞f(q)dq = n

∫ i∞

0

f(q)dq = −nL(E, 1) ∈ L′


lies in the subgroup (L′)+ of elements fixed by complex conjugation. If c is theManin constant, we have cnL(E, 1) ∈ L+. Since ΩE is the least nonzero element ofL+ (or twice it), it follows that 2cnL(E, 1)/ΩE ∈ Z, which proves the proposition.

6Modular Forms of Higher Level

6.1 Modular Forms on Γ1(N)

Fix integers k ≥ 0 and N ≥ 1. Recall that Γ1(N) is the subgroup of elements ofSL2(Z) that are of the form ( 1 ∗

0 1 ) when reduced modulo N .

Definition 6.1.1 (Modular Forms). The space of modular forms of level N andweight k is

Mk(Γ1(N)) =f : f(γτ) = (cτ + d)kf(τ) all γ ∈ Γ1(N)

,

where the f are assumed holomorphic on h ∪ cusps (see below for the precisemeaning of this). The space of cusp forms of level N and weight k is the subspaceSk(Γ1(N)) of Mk(Γ1(N)) of modular forms that vanish at all cusps.

Suppose f ∈Mk(Γ1(N)). The group Γ1(N) contains the matrix ( 1 10 1 ), so

f(z + 1) = f(z),

and for f to be holomorphic at infinity means that f has a Fourier expansion

f =

∞∑

n=0

anqn.

To explain what it means for f to be holomorphic at all cusps, we introducesome additional notation. For α ∈ GL+

2 (R) and f : h→ C define another functionf|[α]k as follows:

f|[α]k(z) = det(α)k−1(cz + d)−kf(αz).

It is straightforward to check that f|[αα′]k = (f|[α]k)|[α′]k . Note that we do not haveto make sense of f|[α]k(∞), since we only assume that f is a function on h andnot h∗.

50 6. Modular Forms of Higher Level

Using our new notation, the transformation condition required for f : h → Cto be a modular form for Γ1(N) of weight k is simply that f be fixed by the [ ]k-action of Γ1(N). Suppose x ∈ P1(Q) is a cusp, and choose α ∈ SL2(Z) such thatα(∞) = x. Then g = f|[α]k is fixed by the [ ]k action of α−1Γ1(N)α.

Lemma 6.1.2. Let α ∈ SL2(Z). Then there exists a positive integer h such that( 1 h

0 1 ) ∈ α−1Γ1(N)α.

Proof. This follows from the general fact that the set of congruence subgroups ofSL2(Z) is closed under conjugation by elements α ∈ SL2(Z), and every congruencesubgroup contains an element of the form ( 1 h

0 1 ). If G is a congruence subgroup,then Γ(N) ⊂ G for some N , and α−1Γ(N)α = Γ(N), since Γ(N) is normal, soΓ(N) ⊂ α−1Gα.

Letting h be as in the lemma, we have g(z + h) = g(z). Then the conditionthat f be holomorphic at the cusp x is that

g(z) =∑

n≥0

bn/hq1/h

on the upper half plane. We say that f vanishes at x if bn/h = 0, so a cusp formis a form that vanishes at every cusp.

6.2 The Diamond Bracket and Hecke Operators

In this section we consider the spaces of modular forms Sk(Γ1(N), ε), for Dirichletcharacters ε mod N , and explicitly describe the action of the Hecke operators onthese spaces.

6.2.1 Diamond Bracket Operators

The group Γ1(N) is a normal subgroup of Γ0(N), and the quotient Γ0(N)/Γ1(N)is isomorphic to (Z/NZ)∗. From this structure we obtain an action of (Z/NZ)∗ onSk(Γ1(N)), and use it to decompose Sk(Γ1(N)) as a direct sum of more manageablechunks Sk(Γ1(N), ε).

Definition 6.2.1 (Dirichlet character). A Dirichlet character ε modulo N isa homomorphism

ε : (Z/NZ)∗ → C∗.

We extend ε to a map ε : Z → C by setting ε(m) = 0 if (m,N) 6= 1 andε(m) = ε(m mod N) otherwise. If ε : Z→ C is a Dirichlet character, the conductorof ε is the smallest positive integer N such that ε arises from a homomorphism(Z/NZ)∗ → C∗.

Remarks 6.2.2.

1. If ε is a Dirichlet character moduloN andM is a multiple ofN then ε inducesa Dirichlet character mod M . If M is a divisor of N then ε is induced by aDirichlet character modulo M if and only if M divides the conductor of ε.

6.2 The Diamond Bracket and Hecke Operators 51

2. The set of Dirichlet characters forms a group, which is non-canonically iso-morphic to (Z/NZ)∗ (it is the dual of this group).

3. The mod N Dirichlet characters all take values in Q(e2πi/e) where e is theexponent of (Z/NZ)∗. When N is an odd prime power, the group (Z/NZ)∗

is cyclic, so e = ϕ(ϕ(N)). This double-ϕ can sometimes cause confusion.

4. There are many ways to represent Dirichlet characters with a computer. Ithink the best way is also the simplest—fix generators for (Z/NZ)∗ in anyway you like and represent ε by the images of each of these generators. As-sume for the moment thatN is odd. To make the representation more “canon-ical”, reduce to the prime power case by writing (Z/NZ)∗ as a product ofcyclic groups corresponding to prime divisors of N . A “canonical” generatorfor (Z/prZ)∗ is then the smallest positive integer s such that s mod pr gen-erates (Z/prZ)∗. Store the character that sends s to e2πin/ϕ(ϕ(pr)) by storingthe integer n. For general N , store the list of integers np, one p for each primedivisor of N (unless p = 2, in which case you store two integers n2 and n′

2,where n2 ∈ 0, 1).

Definition 6.2.3. Let d ∈ (Z/NZ)∗ and f ∈ Sk(Γ1(N)). The map SL2(Z) →SL2(Z/NZ) is surjective, so there exists a matrix γ =

(a bc d

)∈ Γ0(N) such that

d ≡ d (mod N). The diamond bracket d operator is then

f(τ)|〈d〉 = f|[γ]k = f(γτ)(cτ + d)−k.

Remark 6.2.4. Fred Diamond was named after diamond bracket operators.

The definition of 〈d〉 does not depend on the choice of lift matrix(a bc d

), since

any two lifts differ by an element of Γ(N) and f is fixed by Γ(N) since it is fixedby Γ1(N).

For each Dirichlet character ε mod N let

Sk(Γ1(N), ε) = f : f |〈d〉 = ε(d)f all d ∈ (Z/NZ)∗= f : f|[γ]k = ε(dγ)f all γ ∈ Γ0(N),

where dγ is the lower-left entry of γ.When f ∈ Sk(Γ1(N), ε), we say that f has Dirichlet character ε. In the literature,

sometimes f is said to be of “nebentypus” ε.

Lemma 6.2.5. The operator 〈d〉 on the finite-dimensional vector space Sk(Γ1(N))is diagonalizable.

Proof. There exists n such that I = 〈1〉 = 〈dn〉 = 〈d〉n, so the characteristicpolynomial of 〈d〉 divides the square-free polynomial Xn − 1.

Note that Sk(Γ1(N), ε) is the ε(d) eigenspace of 〈d〉. Thus we have a direct sumdecomposition

Sk(Γ1(N)) =⊕

ε:(Z/NZ)∗→C∗

Sk(Γ1(N), ε).

We have(−1 0

0 −1

)∈ Γ0(N), so if f ∈ Sk(Γ1(N), ε), then

f(τ)(−1)−k = ε(−1)f(τ).

Thus Sk(Γ1(N), ε) = 0, unless ε(−1) = (−1)k, so about half of the direct sum-mands Sk(Γ1(N), ε) vanish.


6.2.2 Hecke Operators on q-expansions

Suppose

f =∞∑

n=1

anqn ∈ Sk(Γ1(N), ε),

and let p be a prime. Then

f |Tp =

∞∑

n=1

anpqn + pk−1ε(p)

∞∑

n=1

anqpn, p - N

∞∑

n=1

anpqn + 0. p | N.

Note that ε(p) = 0 when p | N , so the second part of the formula is redundant.When p | N , Tp is often denoted Up in the literature, but we will not do so

here. Also, the ring T generated by the Hecke operators is commutative, so it isharmless, though potentially confusing, to write Tp(f) instead of f |Tp.

We record the relations

TmTn = Tmn, (m,n) = 1,

Tpk =

(Tp)

k, p | NTpk−1Tp − ε(p)pk−1Tpk−2 , p - N.

WARNING: When p | N , the operator Tp on Sk(Γ1(N), ε) need not be diago-nalizable.

6.3 Old and New Subspaces

Let M and N be positive integers such that M | N and let t | NM . If f(τ) ∈

Sk(Γ1(M)) then f(tτ) ∈ Sk(Γ1(N)). We thus have maps

Sk(Γ1(M))→ Sk(Γ1(N))

for each divisor t | NM . Combining these gives a map

ϕM :⊕

t|(N/M)

Sk(Γ1(M))→ Sk(Γ1(N)).

Definition 6.3.1 (Old Subspace). The old subspace of Sk(Γ1(N)) is the sub-space generated by the images of the ϕM for all M | N with M 6= N .

Definition 6.3.2 (New Subspace). The new subspace of Sk(Γ1(N)) is the com-plement of the old subspace with respect to the Petersson inner product.

Since I haven’t introduced the Petersson inner product yet, note that the newsubspace of Sk(Γ1(N)) is the largest subspace of Sk(Γ1(N)) that is stable under theHecke operators and has trivial intersection with the old subspace of Sk(Γ1(N)).

Definition 6.3.3 (Newform). A newform is an element f of the new subspaceof Sk(Γ1(N)) that is an eigenvector for every Hecke operator, which is normalizedso that the coefficient of q in f is 1.

6.3 Old and New Subspaces 53

If f =∑anq

n is a newform then the coefficient an are algebraic integers, whichhave deep arithmetic significance. For example, when f has weight 2, there is anassociated abelian variety Af over Q of dimension [Q(a1, a2, . . .) : Q] such that∏L(fσ, s) = L(Af , s), where the product is over the Gal(Q/Q)-conjugates of F .

The abelian variety Af was constructed by Shimura as follows. Let J1(N) be theJacobian of the modular curveX1(N). As we will see tomorrow, the ring T of Heckeoperators acts naturally on J1(N). Let If be the kernel of the homomorphismT→ Z[a1, a2, . . .] that sends Tn to an. Then

Af = J1(N)/IfJ1(N).

In the converse direction, it is a deep theorem of Breuil, Conrad, Diamond,Taylor, and Wiles that if E is any elliptic curve over Q, then E is isogenous to Affor some f of level equal to the conductor N of E.

When f has weight greater than 2, Scholl constructs, in an analogous way, aGrothendieck motive (=compatible collection of cohomology groups)Mf attachedto f .

7Newforms and Euler Products

In this chapter we discuss the work of Atkin, Lehner, and W. Li on newforms andtheir associated L-series and Euler products. Then we discuss explicitly how Up, forp | N , acts on old forms, and how Up can fail to be diagonalizable. Then we describea canonical generator for Sk(Γ1(N)) as a free module over TC. Finally, we observethat the subalgebra of TQ generated by Hecke operators Tn with (n,N) = 1 isisomorphic to a product of number fields.

7.1 Atkin, Lehner, Li Theory

The results of [Li75] about newforms are proved using many linear transforma-tions that do not necessarily preserve Sk(Γ1(N), ε). Thus we introduce more gen-eral spaces of cusp forms, which these transformations preserve. These spaces arealso useful because they make precise how the space of cusp forms for the fullcongruence subgroup Γ(N) can be understood in terms of spaces Sk(Γ1(M), ε) forvarious M and ε, which justifies our usual focus on these latter spaces. This sectionfollows [Li75] closely.

Let M and N be positive integers and define

Γ0(M,N) =

(a bc d

)∈ SL2(Z) : M | c,N | b

,

and

Γ(M,N) =

(a bc d

)∈ Γ0(M,N) : a ≡ d ≡ 1 (mod MN)

.

Note that Γ0(M, 1) = Γ0(M) and Γ(M, 1) = Γ1(M). Let Sk(M,N) denote thespace of cusp forms for Γ(M,N).

If ε is a Dirichlet character moduloMN such that ε(−1) = (−1)k, let Sk(M,N, ε)denote the space of all cups forms for Γ(M,N) of weight k and character ε. This

56 7. Newforms and Euler Products

is the space of holomorphic functions f : h → C that satisfy the usual vanishingconditions at the cusps and such that for all

(a bc d

)∈ Γ0(M,N),

f |(a bc d

)= ε(d)f.

We have

Sk(M,N) = ⊕εSk(M,N, ε).

We now introduce operators between various Sk(M,N). Note that, except whenotherwise noted, the notation we use for these operators below is as in [Li75],which conflicts with notation in various other books. When in doubt, check thedefinitions.

Let

f |(a bc d

)(τ) = (ad− bc)k/2(cτ + d)−kf

(aτ + b

cτ + d

).

This is like before, but we omit the weight k from the bar notation, since k will befixed for the whole discussion.

For any d and f ∈ Sk(M,N, ε), define

f |UNd = dk/2−1f∣∣∣(

∑

u mod d

(1 uN0 d

)),

where the sum is over any set u of representatives for the integers modulo d. Notethat the N in the notation is a superscript, not a power of N . Also, let

f |Bd = d−k/2f |(d 00 1

),

and

f |Cd = dk/2f |(

1 00 d

).

In [Li75], Cd is denoted Wd, which would be confusing, since in the literature Wd isusually used to denote a completely different operator (the Atkin-Lehner operator,which is denoted V Md in [Li75]).

Since ( 1 N0 1 ) ∈ Γ(M,N), any f ∈ Sk(M,N, ε) has a Fourier expansion in terms

of powers of qN = q1/N . We have

(∑anq

nN

)|UNd =

∑

n≥1

andqnN ,

(∑anq

nN

)|Bd =

∑

n≥1

anqndN ,

and (∑anq

nN

)|Cd =

∑

n≥1

anqnNd.

The second two equalities are easy to see; for the first, write everything out anduse that for n ≥ 1, the sum

∑u e

2πiun/d is 0 or d if d - n, d | n, respectively.

7.1 Atkin, Lehner, Li Theory 57

The maps Bd and Cd define injective maps between various spaces Sk(M,N, ε).To understand Bd, use the matrix relation

(d 00 1

)(x yz w

)=

(x dy

z/d w

)(d 00 1

),

and a similar one for Cd. If d | N then Bd : Sk(M,N, ε) → Sk(dM,N/d, ε) isan isomorphism, and if d | M , then Cd : Sk(M,N) → Sk(M/d,Nd, ε) is also anisomorphism. In particular, taking d = N , we obtain an isomorphism

BN : Sk(M,N, ε)→ Sk(MN, 1, ε) = Sk(Γ1(MN), ε). (7.1.1)

Putting these maps together allows us to completely understand the cusp formsSk(Γ(N)) in terms of spaces Sk(Γ1(N

2), ε), for all Dirichlet characters ε that arisefrom characters modulo N . (Recall that Γ(N) is the principal congruence subgroupΓ(N) = ker(SL2(Z) → SL2(Z/NZ)). This is because Sk(Γ(N)) is isomorphic tothe direct sum of Sk(N,N, ε), as ε various over all Dirichlet characters modulo N .

For any prime p, the pth Hecke operator on Sk(M,N, ε) is defined by

Tp = UNp + ε(p)pk−1Bp.

Note that Tp = UNp when p | N , since then ε(p) = 0. In terms of Fourier expansions,we have (∑

anqnN

)|Tp =

∑

n≥1

(anp + ε(p)pk−1an/p

)qnN ,

where an/p = 0 if p - n.The operators we have just defined satisfy several commutativity relations. Sup-

pose p and q are prime. Then TpBq = BqTp, TpCq = CqTp, and TpUNq = UNq Tp if

(p, qMN) = 1. Moreover UNd Bd′ = Bd′UNd if (d, d′) = 1.

Remark 7.1.1. Because of these relations, (7.1.1) describe Sk(Γ(N)) as a moduleover the ring generated by Tp for p - N .

Definition 7.1.2 (Old Subspace). The old subspace Sk(M,N, ε)old is the sub-space of Sk(M,N, ε) generated by all f |Bd and g|Ce where f ∈ Sk(M ′, N), g ∈Sk(M,N ′), and M ′, N ′ are proper factors of M , N , respectively, and d | M/M ′,e | N/N ′.

Since Tp commutes withBd and Ce, the Hecke operators Tp all preserve Sk(M,N, ε)old,for p - MN . Also, BN defines an isomorphism

Sk(M,N, ε)old ∼= Sk(MN, 1, ε)old.

Definition 7.1.3 (Petersson Inner Product). If f, g ∈ Sk(Γ(N)), the Peters-son inner product of f and g is

〈f, g〉 =1

[SL2(Z) : Γ(N)]

∫

D

f(z)g(z)yk−2 dx dy,

where D is a fundamental domain for Γ(N) and z = x+ iy.

This Petersson pairing is normalized so that if we consider f and g as elementsof Γ(N ′) for some multiple N ′ of N , then the resulting pairing is the same (sincethe volume of the fundamental domain shrinks by the index).


Proposition 7.1.4 (Petersson). If p - N and f ∈ Sk(Γ1(N), ε), then 〈f |Tp, g〉 =ε(p)〈f, g|Tp〉.Remark 7.1.5. The proposition implies that the Tp, for p - N , are diagonalizable.Be careful, because the Tp, with p | N , need not be diagonalizable.

Definition 7.1.6 (New Subspace). The new subspace Sk(M,N, ε)new is theorthogonal complement of Sk(M,N, ε)old in Sk(M,N, ε) with respect to the Pe-tersson inner product.

Both the old and new subspaces of Sk(M,N, ε) are preserved by the Heckeoperators Tp with (p,NM) = 1.

Remark 7.1.7. Li [Li75] also gives a purely algebraic definition of the new subspaceas the intersection of the kernels of various trace maps from Sk(M,N, ε), whichare obtained by averaging over coset representatives.

Definition 7.1.8 (Newform). A newform f =∑anq

nN ∈ Sk(M,N, ε) is an

element of Sk(M,N, ε)new that is an eigenform for all Tp, for p - NM , and isnormalized so that a1 = 1.

Li introduces the crucial “Atkin-Lehner operator” WMq (denoted VMq in [Li75]),

which plays a key roll in all the proofs, and is defined as follows. For a posi-tive integer M and prime q, let α = ordq(M) and find integers x, y, z such thatq2αx − yMz = qα. Then WM

q is the operator defined by slashing with the ma-

trix

(qαx yMz qα

). Li shows that if f ∈ Sk(M, 1, ε), then f |WM

q |WMq = ε(qα)f , so

WMq is an automorphism. Care must be taken, because the operator WM

q need not

commute with Tp = UNp , when p |M .After proving many technical but elementary lemmas about the operators Bd,

Cd, UNp , Tp, and WM

q , Li uses the lemmas to deduce the following theorems. Theproofs are all elementary, but there is little I can say about them, except that youjust have to read them.

Theorem 7.1.9. Suppose f =∑anq

nN ∈ Sk(M,N, ε) and an = 0 for all n with

(n,K) = 1, where K is a fixed positive integer. Then f ∈ Sk(M,N, ε)old.

From the theorem we see that if f and g are newforms in Sk(M,N, ε), and if forall but finitely many primes p, the Tp eigenvalues of f and g are the same, thenf−g is an old form, so f−g = 0, hence f = g. Thus the eigenspaces correspondingto the systems of Hecke eigenvalues associated to the Tp, with p - MN , each havedimension 1. This is known as “multiplicity one”.

Theorem 7.1.10. Let f =∑anq

nN be a newform in Sk(M,N, ε), p a prime with

(p,MN) = 1, and q |MN a prime. Then

1. f |Tp = apf , f |UNq = aqf , and for all n ≥ 1,

apan = anp + ε(p)pk−1an/p,

aqan = anq.

If L(f, s) =∑n≥1 ann

−s is the Dirichlet series associated to f , then L(f, s)has an Euler product

L(f, s) =∏

q|MN

(1− aqq−s)−1∏

p-MN

(1− app−s + ε(p)pk−1p−2s)−1.

7.2 The Up Operator 59

2. (a) If ε is not a character mod MN/q, then |aq| = q(k−1)/2.

(b) If ε is a character mod MN/q, then aq = 0 if q2 | MN , and a2q =

ε(q)qk−2 if q2 - MN .

7.2 The Up Operator

Let N be a positive integer and M a divisor of N . For each divisor d of N/M wedefine a map

αd : Sk(Γ1(M))→ Sk(Γ1(N)) : f(τ) 7→ f(dτ).

We verify that f(dτ) ∈ Sk(Γ1(N)) as follows. Recall that for γ =(a bc d

), we write

(f |[γ]k)(τ) = det(γ)k−1(cz + d)−kf(γ(τ)).

The transformation condition for f to be in Sk(Γ1(N)) is that f |[γ]k(τ) = f(τ). Letf(τ) ∈ Sk(Γ1(M)) and let ιd =

(d 00 1

). Then f |[ιd]k(τ) = dk−1f(dτ) is a modular

form on Γ1(N) since ι−1d Γ1(M)ιd contains Γ1(N). Moreover, if f is a cusp form

then so is f |[ιd]k.Proposition 7.2.1. If f ∈ Sk(Γ1(M)) is nonzero, then

αd(f) : d | N

M

is linearly independent.

Proof. If the q-expansion of f is∑anq

n, then the q-expansion of αd(f) is∑anq

dn.The matrix of coefficients of the q-expansions of αd(f), for d | (N/M), is uppertriangular. Thus the q-expansions of the αd(f) are linearly independent, hencethe αd(f) are linearly independent, since the map that sends a cusp form to itsq-expansion is linear.

When p | N , we denote by Up the Hecke operator Tp acting on the image spaceSk(Γ1(N)). For clarity, in this section we will denote by Tp,M , the Hecke operatorTp ∈ End(Sk(Γ1(M))). For f =

∑anq

n ∈ Sk(Γ1(N)), we have

f |Up =∑

anpqn.

Suppose f =∑anq

n ∈ Sk(Γ1(M)) is a normalized eigenform for all of the Heckeoperators Tn and 〈n〉, and p is a prime that does not divide M . Then

f |Tp,M = apf and f |〈p〉 = ε(p)f.

Assume N = prM , where r ≥ 1 is an integer. Let

fi(τ) = f(piτ),

so f0, . . . , fr are the images of f under the maps αp0 , . . . , αpr , respectively, andf = f0. We have

f |Tp,M =∑

n≥1

anpqn + ε(p)pk−1

∑anq

pn

= f0|Up + ε(p)pk−1f1,


sof0|Up = f |Tp,M − ε(p)pk−1f1 = apf0 − ε(p)pk−1f1.

Alsof1|Up =

(∑anq

pn)|Up =

∑anq

n = f0.

More generally, for any i ≥ 1, we have fi|Up = fi−1.The operator Up preserves the two dimensional vector space spanned by f0

and f1, and the matrix of Up with respect to the basis f0, f1 is

A =

(ap 1

− ε(p)pk−1 0

),

which has characteristic polynomial

X2 − apX + pk−1ε(p). (7.2.1)

7.2.1 A Connection with Galois Representations

This leads to a striking connection with Galois representations. Let f be a newformand let K = Kf be the field generated over Q by the Fourier coefficients of f . Let` be a prime and λ a prime lying over `. Then Deligne (and Serre, when k = 1)constructed a representation

ρλ : Gal(Q/Q)→ GL(2,Kλ).

If p - N`, then ρλ is unramified at p, so if Frobp ∈ Gal(Q/Q) if a Frobenius element,then ρλ(Frobp) is well defined, up to conjugation. Moreover, one can show that

det(ρλ(Frobp)) = pk−1ε(p), and

tr(ρλ(Frobp)) = ap.

(We will discuss the proof of these relations further in the case k = 2.) Thus thecharacteristic polynomial of ρλ(Frobp) ∈ GL2(Eλ) is

X2 − apX + pk−1ε(p),

which is the same as (7.2.1).

7.2.2 When is Up Semisimple?

Question 7.2.2. Is Up semisimple on the span of f0 and f1?

If the eigenvalues of Up are distinct, then the answer is yes. If the eigenvaluesare the same, then X2 − apX + pk−1ε(p) has discriminant 0, so a2

p = 4pk−1ε(p),hence

ap = 2pk−1

2

√ε(p).

Open Problem 7.2.3. Does there exist an eigenform f =∑anq

n ∈ Sk(Γ1(N))

such that ap = 2pk−1

2

√ε(p)?

It is a curious fact that the Ramanujan conjectures, which were proved byDeligne in 1973, imply that |ap| ≤ 2p(k−1)/2, so the above equality remains taunt-ing. When k = 2, Coleman and Edixhoven proved that |ap| < 2p(k−1)/2.

7.3 The Cusp Forms are Free of Rank One over TC 61

7.2.3 An Example of Non-semisimple Up

Suppose f = f0 is a normalized eigenform. Let W be the space spanned by f0, f1and let V be the space spanned by f0, f1, f2, f3. Then Up acts on V/W by f2 7→ 0and f3 7→ f2. Thus the matrix of the action of Up on V/W is ( 0 1

0 0 ), which isnonzero and nilpotent, hence not semisimple. Since W is invariant under Up thisshows that Up is not semisimple on V , i.e., Up is not diagonalizable.

7.3 The Cusp Forms are Free of Rank One over TC

7.3.1 Level 1

Suppose N = 1, so Γ1(N) = SL2(Z). Using the Petersson inner product, we seethat all the Tn are diagonalizable, so Sk = Sk(Γ1(1)) has a basis

f1, . . . , fd

of normalized eigenforms where d = dimSk. This basis is canonical up to ordering.Let TC = T ⊗C be the ring generated over C by the Hecke operator Tp. Then,having fixed the basis above, there is a canonical map

TC → Cd : T 7→ (λ1, . . . , λd),

where fi|T = λifi. This map is injective and dimTC = d, so the map is anisomorphism of C-vector spaces.

The formv = f1 + · · ·+ fn

generates Sk as a T-module. Note that v is canonical since it does not depend onthe ordering of the fi. Since v corresponds to the vector (1, . . . , 1) and T ∼= Cd

acts on Sk ∼= Cd componentwise, this is just the statement that Cd is generatedby (1, . . . , 1) as a Cd-module.

There is a perfect pairing Sk ×TC → C given by

⟨∑f, Tn

⟩= a1(f |Tn) = an(f),

where an(f) denotes the nth Fourier coefficient of f . Thus we have simultaneously:

1. Sk is free of rank 1 over TC, and

2. Sk ∼= HomC(TC,C) as T-modules.

Combining these two facts yields an isomorphism

TC∼= HomC(TC,C). (7.3.1)

This isomorphism sends an element T ∈ T to the homomorphism

X 7→ 〈v|T,X〉 = a1(v|T |X).

Since the identification Sk = HomC(TC,C) is canonical and since the vector v iscanonical, we see that the isomorphism (7.3.1) is canonical.


Recall that Mk has as basis the set of products Ea4Eb6, where 4a + 6b = k, and

Sk is the subspace of forms where the constant coefficient of their q-expansion is 0.Thus there is a basis of Sk consisting of forms whose q-expansions have coefficientsin Q. Let Sk(Z) = Sk ∩ Z[[q]], be the submodule of Sk generated by cusp formswith Fourier coefficients in Z, and note that Sk(Z)⊗Q ∼= Sk(Q). Also, the explicitformula (

∑anq

n)|Tp =∑anpq

n+pk−1∑anq

np implies that the Hecke algebra Tpreserves Sk(Z).

Proposition 7.3.1. The Fourier coefficients of each fi are totally real algebraicintegers.

Proof. The coefficient an(fi) is the eigenvalue of Tn acting on fi. As observedabove, the Hecke operator Tn preserves Sk(Z), so the matrix [Tn] of Tn with respectto a basis for Sk(Z) has integer entries. The eigenvalues of Tn are algebraic integers,since the characteristic polynomial of [Tn] is monic and has integer coefficients.

The eigenvalues are real since the Hecke operators are self-adjoint with respectto the Petersson inner product.

Remark 7.3.2. A CM field is a quadratic imaginary extension of a totally real field.For example, when n > 2, the field Q(ζn) is a CM field, with totally real subfieldQ(ζn)

+ = Q(ζn + 1/ζn). More generally, one shows that the eigenvalues of anynewform f ∈ Sk(Γ1(N)) generate a totally real or CM field.

Proposition 7.3.3. We have v ∈ Sk(Z).

Proof. This is because v =∑

Tr(Tn)qn, and, as we observed above, there is a basis

so that the matrices Tn have integer coefficients.

Example 7.3.4. When k = 36, we have

v = 3q + 139656q2 − 104875308q3 + 34841262144q4 + 892652054010q5

− 4786530564384q6 + 878422149346056q7 + · · · .

The normalized newforms f1, f2, f3 are

fi = q + aq2 + (−1/72a2 + 2697a+ 478011548)q3 + (a2 − 34359738368)q4

(a2 − 34359738368)q4 + (−69/2a2 + 14141780a+ 1225308030462)q5 + · · · ,

for a each of the three roots ofX3−139656X2−59208339456X−1467625047588864.

7.3.2 General Level

Now we consider the case for general level N . Recall that there are maps

Sk(Γ1(M))→ Sk(Γ1(N)),

for all M dividing N and all divisor d of N/M .The old subspace of Sk(Γ1(N)) is the space generated by all images of these

maps with M |N but M 6= N . The new subspace is the orthogonal complement ofthe old subspace with respect to the Petersson inner product.

There is an algebraic definition of the new subspace. One defines trace maps

Sk(Γ1(N))→ Sk(Γ1(M))

7.4 Decomposing the Anemic Hecke Algebra 63

for all M < N , M | N which are adjoint to the above maps (with respect to thePetersson inner product). Then f is in the new part of Sk(Γ1(N)) if and only if fis in the kernels of all of the trace maps.

It follows from Atkin-Lehner-Li theory that the Tn acts semisimply on the newsubspace Sk(Γ1(M))new for all M ≥ 1, since the common eigenspaces for all Tneach have dimension 1. Thus Sk(Γ1(M))new has a basis of normalized eigenforms.We have a natural map

⊕

M |NSk(Γ1(M))new → Sk(Γ1(N)).

The image in Sk(Γ1(N)) of an eigenform f for some Sk(Γ1(M))new is called anewform of level Mf = M . Note that a newform of level less than N is notnecessarily an eigenform for all of the Hecke operators acting on Sk(Γ1(N)); inparticular, it can fail to be an eigenform for the Tp, for p | N .

Let

v =∑

f

f(qN

Mf ) ∈ Sk(Γ1(N)),

where the sum is taken over all newforms f of weight k and some level M | N . Thisgeneralizes the v constructed above when N = 1 and has many of the same goodproperties. For example, Sk(Γ1(N)) is free of rank 1 over T with basis element v.Moreover, the coefficients of v lie in Z, but to show this we need to know thatSk(Γ1(N)) has a basis whose q-expansions lie in Q[[q]]. This is true, but we willnot prove it here. One way to proceed is to use the Tate curve to construct aq-expansion map H0(X1(N),ΩX1(N)/Q) → Q[[q]], which is compatible with theusual Fourier expansion map.

Example 7.3.5. The space S2(Γ1(22)) has dimension 6. There is a single newformof level 11,

f = q − 2q2 − q3 + 2q4 + q5 + 2q6 − 2q7 + · · · .There are four newforms of level 22, the four Gal(Q/Q)-conjugates of

g = q − ζq2 + (−ζ3 + ζ − 1)q3 + ζ2q4 + (2ζ3 − 2)q5

+ (ζ3 − 2ζ2 + 2 ζ − 1)q6 − 2ζ2q7 + ...

where ζ is a primitive 10th root of unity.

Warning 7.3.6. Let S = S2(Γ0(88)), and let v =∑

Tr(Tn)qn. Then S has dimen-

sion 9, but the Hecke span of v only has dimension 7. Thus the more “canonicallooking” element

∑Tr(Tn)q

n is not a generator for S.

7.4 Decomposing the Anemic Hecke Algebra

We first observe that it make no difference whether or not we include the Diamondbracket operators in the Hecke algebra. Then we note that the Q-algebra generatedby the Hecke operators of index coprime to the level is isomorphic to a product offields corresponding to the Galois conjugacy classes of newforms.

Proposition 7.4.1. The operators 〈d〉 on Sk(Γ1(N)) lie in Z[. . . , Tn, . . .].


Proof. It is enough to show 〈p〉 ∈ Z[. . . , Tn, . . .] for primes p, since each 〈d〉 can bewritten in terms of the 〈p〉. Since p - N , we have that

Tp2 = T 2p − 〈p〉pk−1,

so 〈p〉pk−1 = T 2p −Tp2 . By Dirichlet’s theorem on primes in arithmetic progression

[Lan94, VIII.4], there is another prime q congruent to p mod N . Since pk−1 andqk−1 are relatively prime, there exist integers a and b such that apk−1 +bqk−1 = 1.Then

〈p〉 = 〈p〉(apk−1 + bqk−1) = a(Tp2 − Tp2) + b(Tq

2 − Tq2) ∈ Z[. . . , Tn, . . .].

Let S be a space of cusp forms, such as Sk(Γ1(N)) or Sk(Γ1(N), ε). Let

f1, . . . , fd ∈ S

be representatives for the Galois conjugacy classes of newforms in S of level Nfi

dividing N . For each i, let Ki = Q(. . . , an(fi), . . .) be the field generated by theFourier coefficients of fi.

Definition 7.4.2 (Anemic Hecke Algebra). The anemic Hecke algebra is thesubalgebra

T0 = Z[. . . , Tn, . . . : (n,N) = 1] ⊂ T

of T obtained by adjoining to Z only those Hecke operators Tn with n relativelyprime to N .

Proposition 7.4.3. We have T0 ⊗Q ∼=∏di=1Ki.

The map sends Tn to (an(f1), . . . , an(fd)). The proposition can be proved usingthe discussion above and Atkin-Lehner-Li theory, but we will not give a proof here.

Example 7.4.4.When S = S2(Γ1(22)), then T0 ⊗ Q ∼= Q × Q(ζ10) (see Example 7.3.5). WhenS = S2(Γ0(37)), then T0 ⊗Q ∼= Q×Q.

8Hecke operators as correspondences

Our goal is to view the Hecke operators Tn and 〈d〉 as objects defined over Q thatact in a compatible way on modular forms, modular Jacobians, and homology. Inorder to do this, we will define the Hecke operators as correspondences.

8.1 The Definition

Definition 8.1.1 (Correspondence). Let C1 and C2 be curves. A correspon-dence C1 Ã C2 is a curve C together with nonconstant morphisms α : C → C1

and β : C → C2. We represent a correspondence by a diagram

Cα

ÄÄÄÄÄÄÄÄ β

ÂÂ???

???

C2C1

Given a correspondence C1 Ã C2 the dual correspondence C2 Ã C1 is obtainedby looking at the diagram in a mirror

Cβ

ÄÄÄÄÄÄÄÄ α

ÂÂ???

???

C1C2

In defining Hecke operators, we will focus on the simple case when the modularcurve is X0(N) and Hecke operator is Tp, where p - N . We will view Tp as acorrespondence X0(N) Ã X0(N), so there is a curve C = X0(pN) and maps αand β fitting into a diagram

X0(pN)α

ÄÄÄÄÄÄÄ β

ÂÂ???

??

X0(N).X0(N)

66 8. Hecke operators as correspondences

The maps α and β are degeneracy maps which forget data. To define them, we viewX0(N) as classifying isomorphism classes of pairs (E,C), where E is an ellipticcurve and C is a cyclic subgroup of order N (we will not worry about what happensat the cusps, since any rational map of nonsingular curves extends uniquely to amorphism). Similarly, X0(pN) classifies isomorphism classes of pairs (E,G) whereG = C ⊕ D, C is cyclic of order N and D is cyclic of order p. Note that since(p,N) = 1, the group G is cyclic of order pN and the subgroups C and D areuniquely determined by G. The map α forgets the subgroup D of order p, and βquotients out by D:

α : (E,G) 7→ (E,C) (8.1.1)

β : (E,G) 7→ (E/D, (C +D)/D) (8.1.2)

We translate this into the language of complex analysis by thinking of X0(N)and X0(pN) as quotients of the upper half plane. The first map α corresponds tothe map

Γ0(pN)\h→ Γ0(N)\hinduced by the inclusion Γ0(pN) → Γ0(N). The second map β is constructed bycomposing the isomorphism

Γ0(pN)\h ∼−→(p 00 1

)Γ0(pN)

(p 00 1

)−1

\h (8.1.3)

with the map to Γ0(N)\h induced by the inclusion

(p 00 1

)Γ0(pN)

(p 00 1

)−1

⊂ Γ0(N).

The isomorphism (8.1.3) is induced by z 7→(p 00 1

)z = pz; explicitly, it is

Γ0(pN)z 7→(p 00 1

)Γ0(pN)

(p 00 1

)−1 ( p 00 1

)z.

(Note that this is well-defined.)The maps α and β induce pullback maps on differentials

α∗, β∗ : H0(X0(N),Ω1)→ H0(X0(pN),Ω1).

We can identify S2(Γ0(N)) with H0(X0(N),Ω1) by sending the cusp form f(z) tothe holomorphic differential f(z)dz. Doing so, we obtain two maps

α∗, β∗ : S2(Γ0(N))→ S2(Γ0(pN)).

Since α is induced by the identity map on the upper half plane, we have α∗(f) =f , where we view f =

∑anq

n as a cusp form with respect to the smaller groupΓ0(pN). Also, since β∗ is induced by z 7→ pz, we have

β∗(f) = p

∞∑

n=1

anqpn.

The factor of p is because

β∗(f(z)dz) = f(pz)d(pz) = pf(pz)dz.

8.2 Maps induced by correspondences 67

Let X, Y , and C be curves, and α and β be nonconstant holomorphic maps, sowe have a correspondence

Cα

ÄÄÄÄÄÄÄ β

ÂÂ???

??

Y.X

By first pulling back, then pushing forward, we obtain induced maps on differentials

H0(X,Ω1)α∗

−−→ H0(C,Ω1)β∗−→ H0(Y,Ω1).

The composition β∗ α∗ is a map H0(X,Ω1)→ H0(Y,Ω1). If we consider the dualcorrespondence, which is obtained by switching the roles of X and Y , we obtain amap H0(Y,Ω1)→ H0(X,Ω1).

Now let α and β be as in (8.1.1). Then we can recover the action of Tp onmodular forms by considering the induced map

β∗ α∗ : H0(X0(N),Ω1)→ H0(X0(N),Ω1)

and using that S2(Γ0(N)) ∼= H0(X0(N),Ω1).

8.2 Maps induced by correspondences

In this section we will see how correspondences induce maps on divisor groups,which in turn induce maps on Jacobians.

Suppose ϕ : X → Y is a morphism of curves. Let Γ ⊂ X × Y be the graph of ϕ.This gives a correspondence

Γα

ÄÄÄÄÄÄÄ β

ÂÂ???

??

YX

We can reconstruct ϕ from the correspondence by using that ϕ(x) = β(α−1(x)).[draw picture here]

More generally, suppose Γ is a curve and that α : Γ→ X has degree d ≥ 1. Viewα−1(x) as a divisor on Γ (it is the formal sum of the points lying over x, countedwith appropriate multiplicities). Then β(α−1(x)) is a divisor on Y . We thus obtaina map

Divn(X)βα−1

−−−−→ Divdn(Y ),

where Divn(X) is the group of divisors of degree n on X. In particular, settingd = 0, we obtain a map Div0(X)→ Div0(Y ).

We now apply the above construction to Tp. Recall that Tp is the correspondence

X0(pN)α

ÄÄÄÄÄÄÄ β

ÂÂ???

??

X0(N),X0(N)


where α and β are as in Section 8.1 and the induced map is

(E,C)α∗

7→∑

D∈E[p]

(E,C ⊕D)β∗7→

∑

D∈E[p]

(E/D, (C +D)/D).

Thus we have a map Div(X0(N)) → Div(X0(N)). This strongly resembles thefirst definition we gave of Tp on level 1 forms, where Tp was a correspondence oflattices.

8.3 Induced maps on Jacobians of curves

Let X be a curve of genus g over a field k. Recall that there is an importantassociation

curves X/k

−→

Jacobians Jac(X) = J(X) of curves

between curves and their Jacobians.

Definition 8.3.1 (Jacobian). LetX be a curve of genus g over a field k. Then theJacobian of X is an abelian variety of dimension g over k whose underlying groupis functorially isomorphic to the group of divisors of degree 0 on X modulo linearequivalence. (For a more precise definition, see Section ?? (Jacobians section).)

There are many constructions of the Jacobian of a curve. We first consider theAlbanese construction. Recall that over C, any abelian variety is isomorphic toCg/L, where L is a lattice (and hence a free Z-module of rank 2g). There is anembedding

ι : H1(X,Z) → H0(X,Ω1)∗

γ 7→∫

γ

•

Then we realize Jac(X) as a quotient

Jac(X) = H0(X,Ω1)∗/ι(H1(X,Z)).

In this construction, Jac(X) is most naturally viewed as covariantly associatedto X, in the sense that if X → Y is a morphism of curves, then the resulting mapH0(X,Ω1)∗ → H0(Y,Ω1)∗ on tangent spaces induces a map Jac(X)→ Jac(Y ).

There are other constructions in which Jac(X) is contravariantly associatedto X. For example, if we view Jac(X) as Pic0(X), and X → Y is a morphism, thenpullback of divisor classes induces a map Jac(Y ) = Pic0(Y )→ Pic0(X) = Jac(X).

If F : X Ã Y is a correspondence, then F induces an a map Jac(X)→ Jac(Y )and also a map Jac(Y ) → Jac(X). If X = Y , so that X and Y are the same, itcan often be confusing to decide which duality to use. Fortunately, for Tp, with pprime to N , it does not matter which choice we make. But it matters a lot if p | Nsince then we have non-commuting confusable operators and this has resulted inmistakes in the literature.

8.4 More on Hecke operators 69

8.4 More on Hecke operators

Our goal is to move things down to Q from C or Q. In doing this we want tounderstand Tn (or Tp), that is, how they act on the associated Jacobians andhow they can be viewed as correspondences. In characteristic p the formulas ofEichler-Shimura will play an important role.

We consider Tp as a correspondence on X1(N) or X0(N). To avoid confusionwe will mainly consider Tp on X0(N) with p - N . Thus assume, unless otherwisestated, that p - N . Remember that Tp was defined to be the correspondence

X0(pN)α

ÄÄÄÄÄÄÄ β

ÂÂ???

??

X0(N)X0(N)

Think of X0(pN) as consisting of pairs (E,D) where D is a cyclic subgroup of E oforder p and E is the enhanced elliptic curve consisting of an elliptic curve E alongwith a cyclic subgroup of order N . The degeneracy map α forgets the subgroup Dand the degeneracy map β divides by it. By contravariant functoriality we have acommutative diagram

H0(X0(N),Ω1)T∗

p =α∗β∗

// H0(X0(N),Ω1)

S2(Γ0(N))Tp // S2(Γ0(N))

Our convention to define T ∗p as α∗ β∗ instead of β∗ α∗ was completely psy-

chological because there is a canonical duality relating the two. We chose the waywe did because of the analogy with the case of a morphism ϕ : Y → X with graphΓ which induces a correspondence

Γπ1

ÄÄÄÄÄÄÄ π2

ÂÂ???

??

XY

Since the morphism ϕ induces a map on global sections in the other direction

H0(X,Ω1) = Γ(X)ϕ∗

−−→ Γ(Y ) = H0(Y,Ω1)

it is psychologically natural for more general correspondence such as Tp to mapfrom the right to the left.

The morphisms α and β in the definition of Tp are defined over Q. This can beseen using the general theory of representable functors. Thus since Tp is definedover Q most of the algebraic geometric objects we will construct related to Tp willbe defined over Q.

8.5 Hecke operators acting on Jacobians

The Jacobian J(X0(N)) = J0(N) is an abelian variety defined over Q. Thereare both covariant and contravariant ways to construct J0(N). Thus a map α :


X0(pN)→ X0(N) induces maps

J0(pN) J0(pN)

α∗

²²J0(N)

α∗

OO

p+1// J0(N)

Note that α∗ α∗ : J0(N)→ J0(N) is just multiplication by deg(α) = p+ 1, sincethere are p + 1 subgroups of order p in E. (At least when p - N , when p|N thereare only p subgroups.)

There are two possible ways to define Tp as an endomorphism of J0(N). Wecould either define Tp as β∗ α∗ or equivalently as α∗ β∗ (assuming still thatp - N).

8.5.1 The Albanese Map

There is a way to map the curve X0(N) into its Jacobian since the underlyinggroup structure of J0(N) is

J0(N) =

divisors of degree 0 on X0(N)

principal divisors

Once we have chosen a rational point, say ∞, on X0(N) we obtain the Albanesemap

θ : X0(N)→ J0(N) : x 7→ x−∞

which sends a point x to the divisor x−∞. The map θ gives us a way to pullbackdifferentials on J0(N). Let Cot J0(N) denote the cotangent space of J0(N) (or thespace of regular differentials). The diagram

Cot J0(N)

oθ∗

²²

Cot J0(N)ξ∗poo

θ∗o²²

H0(X0(N),Ω1) H0(X0(N),Ω1)T∗

poo

may be taken to give a definition of ξp since there is a unique endomorphismξp : J0(N)→ J0(N) inducing a map ξ∗p which makes the diagram commute.

Now suppose Γ is a correspondence X Ã Y so we have a diagram

Γα

ÄÄÄÄÄÄÄ β

ÂÂ???

??

YX

For example, think of Γ as the graph of a morphism ϕ : X → Y . Then Γ shouldinduce a natural map

H0(Y,Ω1) −→ H0(X,Ω1).

8.5 Hecke operators acting on Jacobians 71

Taking Jacobians we see that the composition

J(X)α∗

−−→ J(Γ)β∗−→ J(Y )

gives a map β∗ α∗ : J(X)→ J(Y ). On cotangent spaces this induces a map

α∗ β∗ : H0(Y,Ω1)→ H0(X,Ω1).

Now, after choice of a rational point, the map X → J(X) induces a mapCot J(X)→ H0(X,Ω1). This is in fact independent of the choice of rational pointsince differentials on J(X) are invariant under translation.

The map J(X)→ J(Y ) is preferred in the literature. It is said to be induced bythe Albanese functoriality of the Jacobian. We could have just as easily defined amap from J(Y )→ J(X). To see this let

ψ = β∗ α∗ : J(X)→ J(Y ).

Dualizing induces a map ψ∨ = α∗ β∗:

J(X)∨

∼=²²

J(Y )∨ψ∨

oo

J(X) J(Y )

∼=

OO

Here we have used autoduality of Jacobians. This canonical duality is discussed in[MFK94] and [Mum70] and in Milne’s article in [Sch65].

8.5.2 The Hecke Algebra

We now have ξp = Tp ∈ End J0(N) for every prime p. If p|N , then we must decidebetween α∗ β∗ and β∗ α∗. The usual choice is the one which induces the usual Tpon cusp forms. If you don’t like your choice you can get out of it with Atkin-Lehneroperators.

LetT = Z[. . . , Tp, . . .] ⊂ End J0(N)

then T is the same as TZ ⊂ End(S2(Γ0(N))). To see this first note that thereis a map T → TZ which is not a prior injective, but which is injective becauseelements of End J0(N) are completely determined by their action on Cot J0(N).


PSfrag replacements

X0(N)

X0(N)

FIGURE 8.6.1. The reduction mod p of the Deligne-Rapoport model of X0(Np)

8.6 The Eichler-Shimura Relation

Suppose p - N is a prime. The Hecke operator Tp and the Frobenius automorphismFrobp induce, by functoriality, elements of End(J0(N)Fp

), which we also denoteTp and Frobp. The Eichler-Shimura relation asserts that the relation

Tp = Frobp +pFrob−1p (8.6.1)

holds in End(J0(N)Fp). In this section we sketch the main idea behind why (8.6.1)

holds. For more details and a proof of the analogous statement for J1(N), see[Con01].

Since J0(N) is an abelian variety defined over Q, it is natural to ask for theprimes p such that J0(N) have good reduction. In the 1950s Igusa showed thatJ0(N) has good reduction for all p - N . He viewed J0(N) as a scheme over Spec(Q),then “spread things out” to make an abelian scheme over Spec(Z[1/N ]). He didthis by taking the Jacobian of the normalization of X0(N) (which is defined overZ[1/N ]) in Pn

Z[1/N ].The Eichler-Shimura relation is a formula for Tp in characteristic p, or more

precisely, for the corresponding endomorphisms in End(J0(N)Fp)) for all p for

which J0(N) has good reduction at p. If p - N , then X0(N)Fphas many of the

same properties as X0(N)Q. In particular, the noncuspidal points on X0(N)Fp

classify isomorphism classes of enhanced elliptic curves E = (E,C), where E is anelliptic curve over Fp and C is a cyclic subgroup of E of order N . (Note that twopairs are considered isomorphic if they are isomorphic over Fp.)

Next we ask what happens to the map Tp : J0(N) → J0(N) under reductionmodulo p. To this end, consider the correspondence

X0(Np)α

ÄÄÄÄÄÄÄ β

ÂÂ???

??

X0(N)X0(N)

that defines Tp. The curve X0(N) has good reduction at p, but X0(Np) typicallydoes not. Deligne and Rapaport [DR73] showed that X0(Np) has relatively benignreduction at p. Over Fp, the reduction X0(Np)Fp

can be viewed as two copies ofX0(N) glued at the supersingular points, as illustrated in Figure 8.6.1.

The set of supersingular points

Σ ⊂ X0(N)(Fp)

is the set of points in X0(N) represented by pairs E = (E,C), where E is asupersingular elliptic curve (so E(Fp)[p] = 0). There are exactly g+1 supersingularpoints, where g is the genus of X0(N).

8.6 The Eichler-Shimura Relation 73

Consider the correspondence Tp : X0(N) Ã X0(N) which takes an enhancedelliptic curve E to the sum

∑E/D of all quotients of E by subgroups D of order p.

This is the correspondence

X0(pN)α

ÄÄÄÄÄÄÄ β

ÂÂ???

??

X0(N),X0(N)

(8.6.2)

where the map α forgets the subgroup of order p, and β quotients out by it. Fromthis one gets Tp : J0(N)→ J0(N) by functoriality.

Remark 8.6.1. There are many ways to think of J0(N). The cotangent spaceCot J0(N) of J0(N) is the space of holomorphic (or translation invariant) dif-ferentials on J0(N), which is isomorphic to S2(Γ0(N)). This gives a connectionbetween our geometric definition of Tp and the definition, presented earlier, of Tpas an operator on a space of cusp forms.

The Eichler-Shimura relation takes place in End(J0(N)Fp). Since X0(N) reduces

“nicely” in characteristic p, we can apply the Jacobian construction to X0(N)Fp.

Lemma 8.6.2. The natural reduction map

End(J0(N)) → End(J0(N)Fp)

is injective.

Proof. Let ` - Np be a prime. By [ST68, Thm. 1, Lem. 2], the reduction to char-acteristic p map induces an isomorphism

J0(N)(Q)[`∞] ∼= J0(N)(Fp)[`∞].

If ϕ ∈ End(J0(N)) reduces to the 0 map in End(J0(N)Fp), then J0(N)(Q)[`∞]

must be contained in ker(ϕ). Thus ϕ induces the 0 map on Tate`(J0(N)), soϕ = 0.

Let F : X0(N)Fp→ X0(N)Fp

be the Frobenius map in characteristic p. Thus,if K = K(X0(N)) is the function field of the nonsingular curve X0(N), thenF : K → K is induced by the pth power map a 7→ ap.

Remark 8.6.3. The Frobenius map corresponds to the pth powering map on points.For example, if X = Spec(Fp[t]), and z = (Spec(Fp)→ X) is a point defined by ahomomorphism α : Fp[t] 7→ Fp, then F (z) is the composite

Fp[t]x7→xp

−−−−−−−→ Fp[t]α−−−−→ Fp.

If α(t) = ξ, then F (z)(t) = α(tp) = ξp.

By both functorialities, F induces maps on the Jacobian of X0(N)Fp:

Frobp = F∗ and Verp = Frob∨p = F ∗,

which we illustrate as follows:

J0(N)Fp

Verp

++J0(N)Fp

Frobp

kk


Note that Verp Frobp = Frobp Verp = [p] since p is the degree of F (for example,if K = Fp(t), then F (K) = Fp(t

p) is a subfield of degree p, so the map inducedby F has degree p).

Theorem 8.6.4 (Eichler-Shimura Relation). Let N be a positive integer andp - N be a prime. Then the following relation holds:

Tp = Frobp +Verp ∈ End(J0(N)Fp).

Sketch of Proof. We view X0(pN)Fpas two copies of X0(N)Fp

glued along corre-sponding supersingular points Σ, as in Figure 8.6.1. This diagram and the corre-spondence (8.6.2) that defines Tp translate into the following diagram of schemesover Fp:

ΣJ j

wwoooooooooooo

· t

''NNNN

NNNN

NNNN

X0(N)Fp · t

r

''NNNN

NNNN

NNN

∼=

²²

X0(N)FpK k

s

xxqqqqqqqqqqq

∼=

²²

X0(pN)Fp

α

wwppppppppppp

β

&&MMMM

MMMM

MMM

X0(N)FpX0(N)Fp

The maps r and s are defined as follows. Recall that a point of X0(N)Fpis an

enhanced elliptic curve E = (E,C) consisting of an elliptic curve E (not necessarilydefined over Fp) along with a cyclic subgroup C of order N . We view a point onX0(Np) as a triple (E,C,E → E′), where (E,C) is as above and E → E ′ isan isogeny of degree p. We use an isogeny instead of a cyclic subgroup of order pbecause E(Fp)[p] has order either 1 or p, so the data of a cyclic subgroup of order pholds very little information.

The map r sends E to (E,ϕ), where ϕ is the isogeny of degree p,

ϕ : EFrob−−−→ E(p).

Here E(p) is the curve obtained from E by hitting all defining equations by Frobe-nious, that is, by pth powering the coefficients of the defining equations for E. Weintroduce E(p) since if E is not defined over Fp, then Frobenious does not definean endomorphism of E. Thus r is the map

r : E 7→ (E,EFrobp−−−→ E(p)),

and similarly we define s to be the map

s : E 7→ (E(p), C,EVerp←−−− E(p))

where Verp is the dual of Frobp (so Verp Frobp = Frobp Verp = [p]).We view α as the map sending (E,E → E ′) to E, and similarly we view β as

the map sending (E,E → E′) to the pair (E′, C ′), where C ′ is the image of C in

8.7 Applications of the Eichler-Shimura Relation 75

E′ via E → E′. Thus

α : (E → E′) 7→ E

β : (E′ → E) 7→ E′

It now follows immediately that α r = id and β s = id. Note also that α s =β r = F is the map E 7→ E(p).

Away from the finitely many supersingular points, we may view X0(pN)Fpas

the disjoint union of two copies of X0(N)Fp. Thus away from the supersingular

points, we have the following equality of correspondences:

X0(pN)Fp

α

ÄÄÄÄÄÄÄ β

ÂÂ???

??

X0(N)FpX0(N)Fp

=′X0(N)Fp

id=αrÄÄÄÄÄÄÄ F=βr

ÂÂ???

??

X0(N)FpX0(N)Fp

+X0(N)Fp

F=αsÄÄÄÄÄÄÄ id=βs

ÂÂ???

??

X0(N)Fp,X0(N)Fp

where F = Frobp, and the =′ means equality away from the supersingular points.Note that we are simply “pulling back” the correspondence; in the first summandwe use the inclusion r, and in the second we use the inclusion s.

This equality of correspondences implies that the equality

Tp = Frobp +Verp

of endomorphisms holds on a dense subset of J0(N)Fp, hence on all J0(N)Fp

.

8.7 Applications of the Eichler-Shimura Relation

8.7.1 The Characteristic Polynomial of Frobenius

How can we apply the relation Tp = Frob+Ver in End(J0(N)Fp)? Let ` - pN be

a prime and consider the `-adic Tate module

Tate`(J0(N)) =(lim←− J0(N)[`ν ]

)⊗Z`

Q`

which is a vector space of dimension 2g over Q`, where g is the genus of X0(N) orthe dimension of J0(N). Reduction modulo p induces an isomorphism

Tate`(J0(N))→ Tate`(J0(N)Fp)

(see the proof of Lemma 8.6.2). On Tate`(J0(N)Fp) we have linear operators Frobp,

Verp and Tp which, as we saw in Section 8.6, satisfy

Frobp +Verp = Tp, and

Frobp Verp = Verp Frobp = [p].

The endomorphism [p] is invertible on Tate`(J0(N)Fp), since p is prime to `, so

Verp and Frobp are also invertible and

Tp = Frobp +[p] Frob−1p .


Multiplying both sides by Frobp and rearranging, we see that

Frob2p−Tp Frobp +[p] = 0 ∈ End(Tate`(J0(N)Fp

)).

This is a beautiful quadratic relation, so we should be able to get something outof it. We will come back to this shortly, but first we consider the various objectsacting on the `-adic Tate module.

The module Tate`(J0(N)) is acted upon in a natural way by

1. The Galois group Gal(Q/Q) of Q, and

2. EndQ(J0(N))⊗Z`Q` (which acts by functoriality).

These actions commute with each other since endomorphisms defined over Q arenot affected by the action of Gal(Q/Q). Reducing modulo p, we also have thefollowing commuting actions:

3. The Galois group Gal(Fp/Fp) of Fp, and

4. EndFp(J0(N))⊗Z`

Q`.

Note that a decomposition group group Dp ⊂ Gal(Q/Q) acts, after quotientingout by the corresponding inertia group, in the same way as Gal(Fp/Fp) and theaction is unramified, so action 3 is a special case of action 1.

The Frobenius elements ϕp ∈ Gal(Fp/Fp) and Frob∈ EndFp(J0(N)) ⊗Z`

Q`

induce the same operator on Tate`(J0(N)Fp). Note that while ϕp naturally lives

in a quotient of a decomposition group, one often takes a lift to get an element inGal(Q/Q).

On Tate`(J0(N)Fp) we have a quadratic relationship

ϕ2p − Tpϕp + p = 0.

This relation plays a role when one separates out pieces of J0(N) in order toconstruct Galois representations attached to newforms of weight 2. Let

R = Z[. . . , Tp, . . .] ⊂ End J0(N),

where we only adjoin those Tp with p - N . Think of R as a reduced Hecke algebra;in particular, R is a subring of T. Then

R⊗Q =

r∏

i=1

Ei,

where the Ei are totally real number fields. The factors Ei are in bijection withthe Galois conjugacy classes of weight 2 newforms f on Γ0(M) (for some M |N).The bijection is the map

f 7→ Q(coefficients of f) = Ei

Observe that the map is the same if we replace f by one of its conjugates. Thisdecomposition is a decomposition of a subring

R⊗Q ⊂ End(J0(N))⊗Qdef= End(J0(N)⊗Q).

8.7 Applications of the Eichler-Shimura Relation 77

Thus it induces a direct product decomposition of J0(N), so J0(N) gets dividedup into subvarieties which correspond to conjugacy classes of newforms.

The relationshipϕ2p − Tpϕp + p = 0 (8.7.1)

suggests thattr(ϕp) = Tp and detϕp = p. (8.7.2)

This is true, but (8.7.2) does not follow formally just from the given quadraticrelation. It can be proved by combining (8.7.1) with the Weil pairing.

8.7.2 The Cardinality of J0(N)(Fp)

Proposition 8.7.1. Let p - N be a prime, and let f be the characteristic polyno-mial of Tp acting on S2(Γ0(N)). Then

#J0(N)(Fp) = f(p+ 1).

9Abelian Varieties

This chapter provides foundational background about abelian varieties and Ja-cobians, with an aim toward what we will need later when we construct abelianvarieties attached to modular forms. We will not give complete proofs of very much,but will try to give precise references whenever possible, and many examples.

We will follow the articles by Rosen [Ros86] and Milne [Mil86] on abelian va-rieties. We will try primarily to explain the statements of the main results aboutabelian varieties, and prove results when the proofs are not too technical andenhance understanding of the statements.

9.1 Abelian Varieties

Definition 9.1.1 (Variety). A variety X over a field k is a finite-type separatedscheme over k that is geometrically integral.

The condition that X be geometrically integral means that Xk is reduced (nonilpotents in the structure sheaf) and irreducible.

Definition 9.1.2 (Group variety). A group variety is a group object in thecategory of varieties. More precisely, a group variety X over a field k is a varietyequipped with morphisms

m : X ×X → X and i : X → X

and a point 1X ∈ A(k) such that m, i, and 1X satisfy the axioms of a group; inparticular, for every k-algebra R they give X(R) a group structure that dependsin a functorial way on R.

Definition 9.1.3 (Abelian Variety). An abelian variety A over a field k is acomplete group variety.

Theorem 9.1.4. Suppose A is an abelian variety. Then

80 9. Abelian Varieties

1. The group law on A is commutative.

2. A is projective, i.e., there is an embedding from A into Pn for some n.

3. If k = C, then A(k) is analytically isomorphic to V/L, where V is a finite-dimensional complex vector space and L is a lattice in V . (A lattice is a freeZ-module of rank equal to 2 dimV such that RL = V .)

Proof. Part 1 is not too difficult, and can be proved by showing that every mor-phism of abelian varieties is the composition of a homomorphism with a transla-tion, then applying this result to the inversion map (see [Mil86, Cor. 2.4]). Part 2is proved with some effort in [Mil86, §7]. Part 3 is proved in [Mum70, §I.1] usingthe exponential map from Lie theory from the tangent space at 0 to A.

9.2 Complex Tori

Let A be an abelian variety over C. By Theorem 9.1.4, there is a complex vectorspace V and a lattice L in V such that A(C) = V/L, that is to say, A(C) is acomplex torus.

More generally, if V is any complex vector space and L is a lattice in V , we callthe quotient T = V/L a complex torus. In this section, we prove some results aboutcomplex tori that will help us to understand the structure of abelian varieties, andwill also be useful in designing algorithms for computing with abelian varieties.

The differential 1-forms and first homology of a complex torus are easy to un-derstand in terms of T . If T = V/L is a complex torus, the tangent space to0 ∈ T is canonically isomorphic to V . The C-linear dual V ∗ = HomC(V,C) isisomorphic to the C-vector space Ω(T ) of holomorphic differential 1-forms on T .Since V → T is the universal covering of T , the first homology H1(T,Z) of T iscanonically isomorphic to L.

9.2.1 Homomorphisms

Suppose T1 = V1/L1 and T2 = V2/L2 are two complex tori. If ϕ : T1 → T2 is a(holomorphic) homomorphism, then ϕ induces a C-linear map from the tangentspace of T1 at 0 to the tangent space of T2 at 0. The tangent space of Ti at 0 iscanonically isomorphic to Vi, so ϕ induces a C-linear map V1 → V2. This maps

9.2 Complex Tori 81

sends L1 into L2, since Li = H1(Ti,Z). We thus have the following diagram:

0

²²

0

²²L1

ρZ(ϕ) //

²²

L2

²²V1

ρC(ϕ) //

²²

L2

²²T1

ϕ //

²²

T2

²²0 0

(9.2.1)

We obtain two faithful representations of Hom(T1, T2),

ρC : Hom(T1, T2)→ HomC(V1, V2)

ρZ : Hom(T1, T2)→ HomZ(L1, L2).

Suppose ψ ∈ HomZ(L1, L2). Then ψ = ρZ(ϕ) for some ϕ ∈ Hom(T1, T2) if andonly if there is a complex linear homomorphism f : V1 → V2 whose restriction toL1 is ψ. Note that f = ψ⊗R is uniquely determined by ψ, so ψ arises from someϕ precisely when f is C-linear. This is the case if and only if fJ1 = J2f , whereJn : Vn → Vn is the R-linear map induced by multiplication by i =

√−1 ∈ C.

Example 9.2.1.

1. Suppose L1 = Z + Zi ⊂ V1 = C. Then with respect to the basis 1, i, wehave J1 =

(0 −11 0

). One finds that Hom(T1, T1) is the free Z-module of rank 2

whose image via ρZ is generated by J1 and ( 1 00 1 ). As a ring Hom(T1, T1) is

isomorphic to Z[i].

2. Suppose L1 = Z + Zαi ⊂ V1 = C, with α3 = 2. Then with respect to

the basis 1, αi, we have J1 =(

0 −α1/α 0

). Only the scalar integer matrices

commute with J1.

Proposition 9.2.2. Let T1 and T2 be complex tori. Then Hom(T1, T2) is a freeZ-module of rank at most 4 dimT1 · dimT2.

Proof. The representation ρZ is faithful (injective) because ϕ is determined by itsaction on L1, since L1 spans V1. Thus Hom(T1, T2) is isomorphic to a subgroup ofHomZ(L1, L2) ∼= Zd, where d = 2dimV1 · 2 dimV2.

Lemma 9.2.3. Suppose ϕ : T1 → T2 is a homomorphism of complex tori. Thenthe image of ϕ is a subtorus of T2 and the connected component of ker(ϕ) is asubtorus of T1 that has finite index in ker(ϕ).


Proof. Let W = ker(ρC(ϕ)). Then the following diagram, which is induced by ϕ,has exact rows and columns:

0

²²

0

²²

0

²²0 // L1 ∩W

²²

// L1//

²²

L2//

²²

L2/ϕ(L1) //

²²

0

0 // W

²²

// V1//

²²

V2//

²²

V2/ϕ(V1) //

²²

0

0 // ker(ϕ) // T1//

²²

T2//

²²

T2/ϕ(T1) //

²²

0

0 0 0

Using the snake lemma, we obtain an exact sequence

0→ L1 ∩W →W → ker(ϕ)→ L2/ϕ(L1)→ V2/ϕ(V1)→ T2/ϕ(T1)→ 0.

Note that T2/ϕ(T1) is compact because it is the continuous image of a compactset, so the cokernel of ϕ is a torus (it is given as a quotient of a complex vectorspace by a lattice).

The kernel ker(ϕ) ⊂ T1 is a closed subset of the compact set T1, so is compact.Thus L1∩W is a lattice inW . The map L2/ϕ(L1)→ V2/ϕ(V1) has kernel generatedby the saturation of ϕ(L1) in L2, so it is finite, so the torus W/(L1 ∩W ) has finiteindex in ker(ϕ).

Remark 9.2.4. The category of complex tori is not an abelian category becausekernels need not be in the category.

9.2.2 Isogenies

Definition 9.2.5 (Isogeny). An isogeny ϕ : T1 → T2 of complex tori is a surjec-tive morphism with finite kernel. The degree deg(ϕ) of ϕ is the order of the kernelof ϕ.

Note that deg(ϕ ϕ′) = deg(ϕ) deg(ϕ′).

Lemma 9.2.6. Suppose that ϕ is an isogeny. Then the kernel of ϕ is isomorphicto the cokernel of ρZ(ϕ).

Proof. (This is essentially a special case of Lemma 9.2.3.) Apply the snake lemmato the morphism (9.2.1) of short exact sequences, to obtain a six-term exact se-quence

0→ KL → KV → KT → CL → CV → CT → 0,

where KX and CX are the kernel and cokernel of X1 → X2, for X = L, V, T ,respectively. Since ϕ is an isogeny, the induced map V1 → V2 must be an isomor-phism, since otherwise the kernel would contain a nonzero subspace (modulo alattice), which would be infinite. Thus KV = CV = 0. It follows that KT

∼= CL, asclaimed.

9.2 Complex Tori 83

One consequence of the lemma is that if ϕ is an isogeny, then

deg(ϕ) = [L1 : ρZ(ϕ)(L1)] = |det(ρZ(ϕ))|.

Proposition 9.2.7. Let T be a complex torus of dimension d, and let n be apositive integer. Then multiplication by n, denoted [n], is an isogeny T → T withkernel T [n] ∼= (Z/nZ)2d and degree n2d.

Proof. By Lemma 9.2.6, T [n] is isomorphic to L/nL, where T = V/L. Since L ≈Z2d, the proposition follows.

We can now prove that isogeny is an equivalence relation.

Proposition 9.2.8. Suppose ϕ : T1 → T2 is a degree m isogeny of complex toriof dimension d. Then there is a unique isogeny ϕ : T2 → T1 of degree m2d−1 suchthat ϕ ϕ = ϕ ϕ = [m].

Proof. Since ker(ϕ) ⊂ ker([m]), the map [m] factors through ϕ, so there is amorphism ϕ such that ϕ ϕ = [m]:

T1ϕ //

[m] ÃÃAAA

AAAA

T2

ϕ

²²T1

We have

(ϕ ϕ− [m]) ϕ = ϕ ϕ φ− [m] ϕ = ϕ ϕ φ−ϕ [m] = ϕ (ϕ φ− [m]) = 0.

This implies that ϕ ϕ = [m], since ϕ is surjective. Uniqueness is clear since thedifference of two such morphisms would vanish on the image of ϕ. To see that ϕhas degree m2d−1, we take degrees on both sides of the equation ϕ ϕ = [m].

9.2.3 Endomorphisms

The ring End(T ) = Hom(T, T ) is called the endomorphism ring of the complextorus T . The endomorphism algebra of T is End0(T ) = End(T )⊗Z Q.

Definition 9.2.9 (Characteristic polynomial). The characteristic polynomialof ϕ ∈ End(T ) is the characteristic polynomial of the ρZ(ϕ). Thus the characteristicpolynomial is a monic polynomial of degree 2 dimT .


9.3 Abelian Varieties as Complex Tori

In this section we introduce extra structure on a complex torus T = V/L that willenable us to understand whether or not T is isomorphic to A(C), for some abelianvariety A over C. When dimT = 1, the theory of the Weierstrass ℘ functionimplies that T is always E(C) for some elliptic curve. In contrast, the generictorus of dimension > 1 does not arise from an abelian variety.

In this section we introduce the basic structures on complex tori that are neededto understand which tori arise from abelian varieties, to construct the dual of anabelian variety, to see that End0(A) is a semisimple Q-algebra, and to understandthe polarizations on an abelian variety. For proofs, including extensive motiva-tion from the one-dimensional case, read the beautifully written book [SD74] bySwinnerton-Dyer, and for another survey that strongly influenced the discussionbelow, see Rosen’s [Ros86].

9.3.1 Hermitian and Riemann Forms

Let V be a finite-dimensional complex vector space.

Definition 9.3.1 (Hermitian form). A Hermitian form is a conjugate-symmetricpairing

H : V × V → C

that is C-linear in the first variable and C-antilinear in the second. Thus H isR-bilinear, H(iu, v) = iH(u, v) = H(u, iv), and H(u, v) = H(v, u).

Write H = S + iE, where S,E : V × V → R are real bilinear pairings.

Proposition 9.3.2. Let H, S, and E be as above.

1. We have that S is symmetric, E is antisymmetric, and

S(u, v) = E(iu, v), S(iu, iv) = S(u, v), E(iu, iv) = E(u, v).

2. Conversely, if E is a real-valued antisymmetric bilinear pairing on V suchthat E(iu, iv) = E(u, v), then H(u, v) = E(iu, v) + iE(u, v) is a Hermitianform on V . Thus there is a bijection between the Hermitian forms on V andthe real, antisymmetric bilinear forms E on V such that E(iu, iv) = E(u, v).

Proof. To see that S is symmetric, note that 2S = H+H and H+H is symmetricbecause H is conjugate symmetric. Likewise, E = (H −H)/(2i), so

E(v, u) =1

2i

(H(v, u)−H(v, u)

)=

1

2i

(H(u, v)−H(u, v)

)= −E(u, v),

which implies that E is antisymmetric. To see that S(u, v) = E(iu, v), rewrite bothS(u, v) and E(iu, v) in terms of H and simplify to get an identity. The other twoidentities follow since

H(iu, iv) = iH(u, iv) = iiH(u, v) = H(u, v).

Suppose E : V × V → R is as in the second part of the proposition. Then

H(iu, v) = E(i2u, v) + iE(iu, v) = −E(u, v) + iE(iu, v) = iH(u, v),

9.3 Abelian Varieties as Complex Tori 85

and the other verifications of linearity and antilinearity are similar. For conjugatesymmetry, note that

H(v, u) = E(iv, u) + iE(v, u) = −E(u, iv)− iE(u, v)

= −E(iu,−v)− iE(u, v) = H(u, v).

Note that the set of Hermitian forms is a group under addition.

Definition 9.3.3 (Riemann form). A Riemann form on a complex torus T =V/L is a Hermitian form H on V such that the restriction of E = Im(H) to L isinteger valued. If H(u, u) ≥ 0 for all u ∈ V then H is positive semi-definite and ifH is positive and H(u, u) = 0 if and only if u = 0, then H is nondegenerate.

Theorem 9.3.4. Let T be a complex torus. Then T is isomorphic to A(C), forsome abelian variety A, if and only if there is a nondegenerate Riemann form on T .

This is a nontrivial theorem, which we will not prove here. It is proved in [SD74,Ch.2] by defining an injective map from positive divisors on T = V/L to posi-tive semi-definite Riemann forms, then constructing positive divisors associated totheta functions on V . If H is a nondegenerate Riemann form on T , one computesthe dimension of a space of theta functions that corresponds to H in terms of thedeterminant of E = Im(H). Since H is nondegenerate, this space of theta functionsis nonzero, so there is a corresponding nondegenerate positive divisor D. Then abasis for

L(3D) = f : (f) + 3D is positive ∪ 0determines an embedding of T in a projective space.

Why the divisor 3D instead of D above? For an elliptic curve y2 = x3 + ax+ b,we could take D to be the point at infinity. Then L(3D) consists of the functionswith a pole of order at most 3 at infinity, which contains 1, x, and y, which havepoles of order 0, 2, and 3, respectively.

Remark 9.3.5. (Copied from page 39 of [SD74].) When n = dimV > 1, however,a general lattice L will admit no nonzero Riemann forms. For if λ1, . . . , λ2n is abase for L then E as an R-bilinear alternating form is uniquely determined bythe E(λi, λj), which are integers; and the condition E(z, w) = E(iz, iw) induceslinear relations with real coefficients between E(λi, λj), which for general L haveno nontrivial integer solutions.

9.3.2 Complements, Quotients, and Semisimplicity of the

Endomorphism Algebra

Lemma 9.3.6. If T possesses a nondegenerate Riemann form and T ′ ⊂ T is asubtorus, then T ′ also possesses a nondegenerate Riemann form.

Proof. If H is a nondegenerate Riemann form on a torus T and T ′ is a subtorusof T , then the restriction of H to T ′ is a nondegenerate Riemann form on T ′ (therestriction is still nondegenerate because H is positive definite).

Lemma 9.3.6 and Lemma 9.2.3 together imply that the kernel of a homomor-phism of abelian varieties is an extension of an abelian variety by a finite group.


Lemma 9.3.7. If T possesses a nondegenerate Riemann form and T → T ′ is anisogeny, then T ′ also possesses a nondegenerate Riemann form.

Proof. Suppose T = V/L and T ′ = V ′/L′. Since the isogeny is induced by anisomorphism V → V ′ that sends L into L′, we may assume for simplicity that V =V ′ and L ⊂ L′. If H is a nondegenerate Riemann form on V/L, then E = Re(H)need not be integer valued on L′. However, since L has finite index in L′, thereis some integer d so that dE is integer valued on L′. Then dH is a nondegenerateRiemann form on V/L′.

Note that Lemma 9.3.7 implies that the quotient of an abelian variety by a finitesubgroup is again an abelian variety.

Theorem 9.3.8 (Poincare Reducibility). Let A be an abelian variety and sup-pose A′ ⊂ A is an abelian subvariety. Then there is an abelian variety A′′ ⊂ Asuch that A = A′ +A′′ and A′ ∩A′′ is finite. (Thus A is isogenous to A′ ×A′′.)

Proof. We have A(C) ≈ V/L and there is a nondegenerate Riemann form Hon V/L. The subvariety A′ is isomorphic to V ′/L′, where V ′ is a subspace of Vand L′ = V ′ ∩ L. Let V ′′ be the orthogonal complement of V ′ with respect to H,and let L′′ = L∩V ′′. To see that L′′ is a lattice in V ′′, it suffices to show that L′′ isthe orthogonal complement of L′ in L with respect to E = Im(H), which, becauseE is integer valued, will imply that L′′ has the correct rank. First, suppose thatv ∈ L′′; then, by definition, v is in the orthogonal complement of L′ with respectto H, so for any u ∈ L′, we have 0 = H(u, v) = S(u, v) + iE(u, v), so E(u, v) = 0.Next, suppose that v ∈ L satisfies E(u, v) = 0 for all u ∈ L′. Since V ′ = RL′ and Eis R-bilinear, this implies E(u, v) = 0 for any u ∈ V ′. In particular, since V ′ is acomplex vector space, if u ∈ L′, then S(u, v) = E(iu, v) = 0, so H(u, v) = 0.

We have shown that L′′ is a lattice in V ′′, so A′′ = V ′′/L′′ is an abelian subvarietyof A. Also L′+L′′ has finite index in L, so there is an isogeny V ′/L′⊕V ′′/L′′ → V/Linduced by the natural inclusions.

Proposition 9.3.9. Suppose A′ ⊂ A is an inclusion of abelian varieties. Thenthe quotient A/A′ is an abelian variety.

Proof. Suppose A = V/L and A′ = V ′/L′, where V ′ is a subspace of V . LetW = V/V ′ and M = L/(L ∩ V ′). Then, W/M is isogenous to the complex torusV ′′/L′′ of Theorem 9.3.8 via the natural map V ′′ → W . Applying Lemma 9.3.7completes the proof.

Definition 9.3.10. An abelian variety A is simple if it has no nonzero properabelian subvarieties.

Proposition 9.3.11. The algebra End0(A) is semisimple.

Proof. Using Theorem 9.3.8 and induction, we can find an isogeny

A ' An1

1 ×An2

2 × · · · ×Anrr

with each Ai simple. Since End0(A) = End(A)⊗Q is unchanged by isogeny, andHom(Ai, Aj) = 0 when i 6= j, we have

End0(A) = End0(An1

1 )× End0(An2

2 )× · · · × End0(Anrr )

9.3 Abelian Varieties as Complex Tori 87

Each of End0(Ani

i ) is isomorphic to Mni(Di), where Di = End0(Ai). By Schur’s

Lemma, Di = End0(Ai) is a division algebra over Q (proof: any nonzero endomor-phism has trivial kernel, and any injective linear transformation of a Q-vector spaceis invertible), so End0(A) is a product of matrix algebras over division algebrasover Q, which proves the proposition.

9.3.3 Theta Functions

Suppose T = V/L is a complex torus.

Definition 9.3.12 (Theta function). Let M : V × L → C and J : L → C beset-theoretic maps such that for each λ ∈ L the map z 7→ M(z, λ) is C-linear. Atheta function of type (M,J) is a function θ : V → C such that for all z ∈ V andλ ∈ L, we have

θ(z + λ) = θ(z) · exp(2πi(M(z, λ) + J(λ))).

Suppose that θ(z) is a nonzero holomorphic theta function of type (M,J). TheM(z, λ), for various λ, cannot be unconnected. Let F (z, λ) = 2πi(M(z, λ)+J(λ)).

Lemma 9.3.13. For any λ, λ′ ∈ L, we have

F (z, λ+ λ′) = F (z + λ, λ′) + F (z, λ) (mod 2πi).

ThusM(z, λ+ λ′) = M(z, λ) +M(z, λ′), (9.3.1)

andJ(λ+ λ′)− J(λ)− J(λ′) ≡M(λ, λ′) (mod Z).

Proof. Page 37 of [SD74].

Using (9.3.1) we see that M extends uniquely to a function M : V × V → Cwhich is C-linear in the first argument and R-linear in the second. Let

E(z, w) = M(z, w)−M(w, z),

H(z, w) = E(iz, w) + iE(z, w).

Proposition 9.3.14. The pairing H is Riemann form on T with real part E.

We call H the Riemann form associated to θ.


9.4 A Summary of Duality and Polarizations

Suppose A is an abelian variety over an arbitrary field k. In this section we sum-marize the most important properties of the dual abelian variety A∨ of A. First wereview the language of sheaves on a scheme X, and define the Picard group of X asthe group of invertible sheaves on X. The dual of A is then a variety whose pointscorrespond to elements of the Picard group that are algebraically equivalent to 0.Next, when the ground field is C, we describe how to view A∨ as a complex torusin terms of a description of A as a complex torus. We then define the Neron-Severigroup of A and relate it to polarizations of A, which are certain homomorphismsA→ A∨. Finally we observe that the dual is functorial.

9.4.1 Sheaves

We will use the language of sheaves, as in [Har77], which we now quickly recall. Apre-sheaf of abelian groups F on a scheme X is a contravariant functor from thecategory of open sets on X (morphisms are inclusions) to the category of abeliangroups. Thus for every open set U ⊂ X there is an abelian group F(U), and ifU ⊂ V , then there is a restriction map F(V ) → F(U). (We also require thatF(∅) = 0, and the map F(U)→ F(U) is the identity map.) A sheaf is a pre-sheafwhose sections are determined locally (for details, see [Har77, §II.1]).

Every scheme X is equipped with its structure sheaf OX , which has the propertythat if U = Spec(R) is an affine open subset of X, then OX(U) = R. A sheaf ofOX-modules is a sheafM of abelian groups on X such that each abelian group hasthe structure of OX -module, such that the restriction maps are module morphisms.A locally-free sheaf of OX -modules is a sheafM of OX -modules, such that X canbe covered by open sets U so that M|U is a free OX -module, for each U .

9.4.2 The Picard Group

An invertible sheaf is a sheaf L of OX -modules that is locally free of rank 1. If Lis an invertible sheaf, then the sheaf-theoretic Hom, L∨ = Hom(L,OX) has theproperty that L∨⊗L = OX . The group Pic(X) of invertible sheaves on a scheme Xis called the Picard group of X. See [Har77, §II.6] for more details.

Let A be an abelian variety over a field k. An invertible sheaf L on A is alge-braically equivalent to 0 if there is a connected variety T over k, an invertible sheafM on A×k T , and t0, t1 ∈ T (k) such that Mt0

∼= L and Mt1∼= OA. Let Pic0(A)

be the subgroup of elements of Pic(A) that are algebraically equivalent to 0.The dual A∨ of A is a (unique up to isomorphism) abelian variety such that for

every field F that contains the base field k, we have A∨(F ) = Pic0(AF ). For theprecise definition of A∨ and a proof that A∨ exists, see [Mil86, §9–10].

9.4.3 The Dual as a Complex Torus

When A is defined over the complex numbers, so A(C) = V/L for some vectorspace V and some lattice L, [Ros86, §4] describes a construction of A∨ as a complextorus, which we now describe. Let

V ∗ = f ∈ HomR(V,C) : f(αt) = αf(t), all α ∈ C, t ∈ V .

9.4 A Summary of Duality and Polarizations 89

Then V ∗ is a complex vector space of the same dimension as V and the map〈f, v〉 = Imf(t) is an R-linear pairing V ∗ × V → R. Let

L∗ = f ∈ V ∗ : 〈f, λ〉 ∈ Z, all λ ∈ L.

Since A is an abelian variety, there is a nondegenerate Riemann form H on A.The map λ : V → V ∗ defined by λ(v) = H(v, ·) is an isomorphism of complexvector spaces. If v ∈ L, then λ(v) = H(v, ·) is integer valued on L, so λ(L) ⊂ L∗.Thus λ induces an isogeny of complex tori V/L→ V ∗/L∗, so by Lemma 9.3.7 thetorus V ∗/L∗ possesses a nondegenerate Riemann form (it’s a multiple of H). In[Ros86, §4], Rosen describes an explicit isomorphism between V ∗/L∗ and A∨(C).

9.4.4 The Neron-Several Group and Polarizations

Let A be an abelian variety over a field k. Recall that Pic(A) is the group ofinvertible sheaves on A, and Pic0(A) is the subgroup of invertible sheaves thatare algebraically equivalent to 0. The Neron-Severi group of A is the quotientPic(A)/Pic0(A), so by definition we have an exact sequence

0→ Pic0(A)→ Pic(A)→ NS(A)→ 0.

Suppose L is an invertible sheaf on A. One can show that the map A(k) →Pic0(A) defined by a 7→ t∗aL ⊗ L−1 is induced by homomorphism ϕL : A →A∨. (Here t∗aL is the pullback of the sheaf L by translation by a.) Moreover, themap L 7→ ϕL induces a homomorphism from Pic(A) → Hom(A,A∨) with kernelPic0(A). The group Hom(A,A∨) is free of finite rank, so NS(A) is a free abeliangroup of finite rank. Thus Pic0(A) is saturated in Pic(A) (i.e., the cokernel of theinclusion Pic0(A)→ Pic(A) is torsion free).

Definition 9.4.1 (Polarization). A polarization on A is a homomorphism λ :A→ A∨ such that λk = ϕL for some L ∈ Pic(Ak). A polarization is principal if itis an isomorphism.

When the base field k is algebraically closed, the polarizations are in bijectionwith the elements of NS(A). For example, when dimA = 1, we have NS(A) = Z,and the polarizations on A are multiplication by n, for each integer n.

9.4.5 The Dual is Functorial

The association A 7→ A∨ extends to a contravariant functor on the category ofabelian varieties. Thus if ϕ : A→ B is a homomorphism, there is a natural choiceof homomorphism ϕ∨ : B∨ → A∨. Also, (A∨)∨ = A and (ϕ∨)∨ = ϕ.

Theorem 9.4.2 below describes the kernel of ϕ∨ in terms of the kernel of ϕ.If G is a finite group scheme, the Cartier dual of G is Hom(G,Gm). For example,the Cartier dual of Z/mZ is µm and the Cartier dual of µm is Z/mZ. (If k isalgebraically closed, then the Cartier dual of G is just G again.)

Theorem 9.4.2. If ϕ : A→ B is a surjective homomorphism of abelian varietieswith kernel G, so we have an exact sequence 0 → G → A → B → 0, then thekernel of ϕ∨ is the Cartier dual of G, so we have an exact sequence 0 → G∨ →B∨ → A∨ → 0.


9.5 Jacobians of Curves

We begin this lecture about Jacobians with an inspiring quote of David Mumford:

“The Jacobian has always been a corner-stone in the analysis of alge-braic curves and compact Riemann surfaces. [...] Weil’s construction [ofthe Jacobian] was the basis of his epoch-making proof of the RiemannHypothesis for curves over finite fields, which really put characteris-tic p algebraic geometry on its feet.” – Mumford, Curves and TheirJacobians, page 49.

9.5.1 Divisors on Curves and Linear Equivalence

Let X be a projective nonsingular algebraic curve over an algebraically field k. Adivisor on X is a formal finite Z-linear combination

∑mi=1 niPi of closed points in

X. Let Div(X) be the group of all divisors on X. The degree of a divisor∑mi=1 niPi

is the integer∑mi=1 ni. Let Div0(X) denote the subgroup of divisors of degree 0.

Suppose k is a perfect field (for example, k has characteristic 0 or k is finite),but do not require that k be algebraically closed. Let the group of divisors on Xover k be the subgroup

Div(X) = Div(X/k) = H0(Gal(k/k),Div(X/k))

of elements of Div(X/k) that are fixed by every automorphism of k/k. Likewise,let Div0(X/k) be the elements of Div(X/k) of degree 0.

A rational function on an algebraic curve X is a function X → P1, defined bypolynomials, which has only a finite number of poles. For example, if X is theelliptic curve over k defined by y2 = x3 +ax+b, then the field of rational functionson X is the fraction field of the integral domain k[x, y]/(y2 − (x3 + ax + b)). LetK(X) denote the field of all rational functions on X defined over k.

There is a natural homomorphismK(X)∗ → Div(X) that associates to a rationalfunction f its divisor

(f) =∑

ordP (f) · P

where ordP (f) is the order of vanishing of f at P . Since X is nonsingular, the localring of X at a point P is isomorphic to k[[t]]. Thus we can write f = trg(t) forsome unit g(t) ∈ k[[t]]. Then R = ordP (f).

Example 9.5.1. If X = P1, then the function f = x has divisor (0)− (∞). If X isthe elliptic curve defined by y2 = x3 + ax+ b, then

(x) = (0,√b) + (0,−

√b)− 2∞,

and

(y) = (x1, 0) + (x2, 0) + (x3, 0)− 3∞,where x1, x2, and x3 are the roots of x3+ax+b = 0. A uniformizing parameter t atthe point ∞ is x/y. An equation for the elliptic curve in an affine neighborhood of∞ is Z = X3 + aXZ2 + bZ3 (where ∞ = (0, 0) with respect to these coordinates)and x/y = X in these new coordinates. By repeatedly substituting Z into thisequation we see that Z can be written in terms of X.

9.5 Jacobians of Curves 91

It is a standard fact in the theory of algebraic curves that if f is a nonzero rationalfunction, then (f) ∈ Div0(X), i.e., the number of poles of f equals the number ofzeros of f . For example, if X is the Riemann sphere and f is a polynomial, thenthe number of zeros of f (counted with multiplicity) equals the degree of f , whichequals the order of the pole of f at infinity.

The Picard group Pic(X) of X is the group of divisors on X modulo linearequivalence. Since divisors of functions have degree 0, the subgroup Pic0(X) ofdivisors on X of degree 0, modulo linear equivalence, is well defined. Moreover, wehave an exact sequence of abelian groups

0→ K(X)∗ → Div0(X)→ Pic0(X)→ 0.

Thus for any algebraic curve X we have associated to it an abelian groupPic0(X). Suppose π : X → Y is a morphism of algebraic curves. If D is a di-visor on Y , the pullback π∗(D) is a divisor on X, which is defined as follows.If P ∈ Div(Y/k) is a point, let π∗(P ) be the sum

∑eQ/PQ where π(Q) = P

and eQ/P is the ramification degree of Q/P . (Remark: If t is a uniformizer at Pthen eQ/P = ordQ(φ∗tP ).) One can show that π∗ : Div(Y ) → Div(X) induces

a homomorphism Pic0(Y ) → Pic0(X). Furthermore, we obtain the contravariantPicard functor from the category of algebraic curves over a fixed base field tothe category of abelian groups, which sends X to Pic0(X) and π : X → Y toπ∗ : Pic0(Y )→ Pic0(X).

Alternatively, instead of defining morphisms by pullback of divisors, we couldconsider the push forward. Suppose π : X → Y is a morphism of algebraic curvesand D is a divisor on X. If P ∈ Div(X/k) is a point, let π∗(P ) = π(P ). Then π∗induces a morphism Pic0(X) → Pic0(Y ). We again obtain a functor, called thecovariant Albanese functor from the category of algebraic curves to the categoryof abelian groups, which sends X to Pic0(X) and π : X → Y to π∗ : Pic0(X) →Pic0(Y ).

9.5.2 Algebraic Definition of the Jacobian

First we describe some universal properties of the Jacobian under the hypothesisthat X(k) 6= ∅. Thus suppose X is an algebraic curve over a field k and thatX(k) 6= ∅. The Jacobian variety of X is an abelian variety J such that for anextension k′/k, there is a (functorial) isomorphism J(k′) → Pic0(X/k′). (I don’tknow whether this condition uniquely characterizes the Jacobian.)

Fix a point P ∈ X(k). Then we obtain a map f : X(k)→ Pic0(X/k) by sendingQ ∈ X(k) to the divisor class ofQ−P . One can show that this map is induced by aninjective morphism of algebraic varieties X → J . This morphism has the followinguniversal property: if A is an abelian variety and g : X → A is a morphism thatsends P to 0 ∈ A, then there is a unique homomorphism ψ : J → A of abelianvarieties such that g = ψ f :

Xf //

gÃÃ@

@@@@

@@J

ψ

²²A

This condition uniquely characterizes J , since if f ′ : X → J ′ and J ′ has the univer-sal property, then there are unique maps J → J ′ and J ′ → J whose composition


in both directions must be the identity (use the universal property with A = Jand f = g).

If X is an arbitrary curve over an arbitrary field, the Jacobian is an abelianvariety that represents the “sheafification” of the “relative Picard functor”. Lookin Milne’s article or Bosch-Luktebohmert-Raynaud Neron Models for more details.Knowing this totally general definition won’t be important for this course, sincewe will only consider Jacobians of modular curves, and these curves always havea rational point, so the above properties will be sufficient.

A useful property of Jacobians is that they are canonically principally polarized,by a polarization that arises from the “θ divisor” on J . In particular, there is alwaysan isomorphism J → J∨ = Pic0(J).

9.5.3 The Abel-Jacobi Theorem

Over the complex numbers, the construction of the Jacobian is classical. It wasfirst considered in the 19th century in order to obtain relations between integralsof rational functions over algebraic curves (see Mumford’s book, Curves and TheirJacobians, Ch. III, for a nice discussion).

Let X be a Riemann surface, so X is a one-dimensional complex manifold.Thus there is a system of coordinate charts (Uα, tα), where tα : Uα → C is ahomeomorphism of Uα onto an open subset of C, such that the change of coordinatemaps are analytic isomorphisms. A differential 1-form on X is a choice of twocontinuous functions f and g to each local coordinate z = x + iy on Uα ⊂ Xsuch that f dx+ g dy is invariant under change of coordinates (i.e., if another localcoordinate patch U ′

α intersects Uα, then the differential is unchanged by the changeof coordinate map on the overlap). If γ : [0, 1]→ X is a path and ω = f dx+ g dyis a 1-form, then

∫

γ

ω :=

∫ 1

0

(f(x(t), y(t))

dx

dt+ g(x(t), y(t))

dy

dt

)dt ∈ C.

From complex analysis one sees that if γ is homologous to γ ′, then∫γω =

∫γ′ ω.

In fact, there is a nondegenerate pairing

H0(X,Ω1X)×H1(X,Z)→ C

If X has genus g, then it is a standard fact that the complex vector spaceH0(X,Ω1

X) of holomorphic differentials on X is of dimension g. The integrationpairing defined above induces a homomorphism from integral homology to thedual V of the differentials:

Φ : H1(X,Z)→ V = Hom(H0(X,Ω1X),C).

This homomorphism is called the period mapping.

Theorem 9.5.2 (Abel-Jacobi). The image of Φ is a lattice in V .

The proof involves repeated clever application of the residue theorem.The intersection pairing

H1(X,Z)×H1(X,Z)→ Z

9.5 Jacobians of Curves 93

defines a nondegenerate alternating pairing on L = Φ(H1(X,Z)). This pairingsatisfies the conditions to induce a nondegenerate Riemann form on V , which givesJ = V/L to structure of abelian variety. The abelian variety J is the Jacobian of X,

and if P ∈ X, then the functional ω 7→∫ QPω defines an embedding of X into J .

Also, since the intersection pairing is perfect, it induces an isomorphism from J toJ∨.

Example 9.5.3. For example, suppose X = X0(23) is the modular curve attachedto the subgroup Γ0(23) of matrices in SL2(Z) that are upper triangular modulo 24.Then g = 2, and a basis for H1(X0(23),Z) in terms of modular symbols is

−1/19, 0, −1/17, 0, −1/15, 0, −1/11, 0.

The matrix for the intersection pairing on this basis is

0 −1 −1 −11 0 −1 −11 1 0 −11 1 1 0

With respect to a reduced integral basis for

H0(X,Ω1X) ∼= S2(Γ0(23)),

the lattice Φ(H1(X,Z)) of periods is (approximately) spanned by

[

(0.59153223605591049412844857432 - 1.68745927346801253993135357636*i

0.762806324458047168681080323846571478727 - 0.60368764497868211035115379488*i),

(-0.59153223605591049412844857432 - 1.68745927346801253993135357636*i

-0.762806324458047168681080323846571478727 - 0.60368764497868211035115379488*i),

(-1.354338560513957662809528899804 - 1.0837716284893304295801997808568748714097*i

-0.59153223605591049412844857401 + 0.480083983510648319229045987467*i),

(-1.52561264891609433736216065099 0.342548176804273349105263499648)

]


9.5.4 Every abelian variety is a quotient of a Jacobian

Over an infinite field, every abelin variety can be obtained as a quotient of aJacobian variety. The modular abelian varieties that we will encounter later are,by definition, exactly the quotients of the Jacobian J1(N) of X1(N) for some N .In this section we see that merely being a quotient of a Jacobian does not endowan abelian variety with any special properties.

Theorem 9.5.4 (Matsusaka). Let A be an abelian variety over an algebraicallyclosed field. Then there is a Jacobian J and a surjective map J → A.

This was originally proved in On a generating curve of an abelian variety, Nat.Sc. Rep. Ochanomizu Univ. 3 (1952), 1–4. Here is the Math Review by P. Samuel:

An abelian variety A is said to be generated by a variety V (and amapping f of V into A) if A is the group generated by f(V ). It is provedthat every abelian variety A may be generated by a curve defined overthe algebraic closure of def(A). A first lemma shows that, if a varietyV is the carrier of an algebraic system (X(M))M∈U of curves (X(M)being defined, non-singular and disjoint from the singular bunch of Vfor almost all M in the parametrizing variety U) if this system has asimple base point on V , and if a mapping f of V into an abelian varietyis constant on some X(M0), then f is a constant; this is proved byspecializing on M0 a generic point M of U and by using specializationsof cycles [Matsusaka, Mem. Coll. Sci. Kyoto Univ. Ser. A. Math. 26,167–173 (1951); these Rev. 13, 379]. Another lemma notices that, for anormal projective variety V , a suitable linear family of plane sectionsof V may be taken as a family (X(M)). Then the main result followsfrom the complete reducibility theorem. This result is said to be thebasic tool for generalizing Chow’s theorem (”the Jacobian variety of acurve defined over k is an abelian projective variety defined over k”).

Milne [Mil86, §10] proves the theorem under the weaker hypothesis that the basefield is infinite. We briefly sketch his proof now. If dimA = 1, then A is theJacobian of itself, so we may assume dimA > 1. Embed A into Pn, then, usingthe Bertini theorem, cut A ⊂ Pn by hyperplane sections dim(A) − 1 times toobtain a nonsingular curve C on A of the form A∩V , where V is a linear subspaceof Pn. Using standard arguments from Hartshorne [Har77], Milne shows (Lemma10.3) that if W is a nonsingular variety and π : W → A is a finite morphism, thenπ−1(C) is geometrically connected (the main point is that the pullback of an ampleinvertible sheaf by a finite morphism is ample). (A morphism f : X → Y is finiteif for every open affine subset U = Spec(R) ⊂ Y , the inverse image f−1(U) ⊂ Xis an affine open subset Spec(B) with B a finitely generated R-module. Finitemorphisms have finite fibers, but not conversely.) We assume this lemma anddeduce the theorem.

Let J be the Jacobian of C; by the universal property of Jacobians there isa unique homomorphism f : J → A coming from the inclusion C → A. Theimage A1 = f(J) is an abelian subvariety since images of homomorphisms ofabelian varieties are abelian varieties. By the Poincare reducibility theorem (weonly proved this over C, but it is true in general), there is an abelian subvarietyA2 ⊂ A such that A1 +A2 = A and A1∩A2 is finite. The isogeny g : A1×A2 → Agiven by g(x, y) = x+ y ∈ A is a finite morphism (any isogeny of abelian varieties

9.6 Neron Models 95

is finite, flat, and surjective by Section 8 of [Mil86]). The inverse image g−1(A1)is a union of #(A1 ∩A2) irreducible components; if this intersection is nontrivial,then likewise g−1(C) is reducible, which is a contradiction. This does not completethe proof, since it is possible that g is an isomorphism, so we use one additionaltrick. Suppose n is a positive integer coprime to the residue characteristic, and let

h = 1× [n] : A1 ×A2 → A1 ×A2

be the identity map on the first factor and multiplication by n on the second.Then h is finite and (h g)−1(A1) is a union of n2 dimA2 = deg(h) irreduciblecomponents, hence (h g)−1(C) is reducible, a contradiction.

Question 9.5.5. Is Theorem 9.5.4 false for some abelian variety A over somefinite field k?

Question 9.5.6 (Milne). Using the theorem we can obtain a sequence of Jaco-bian varieties J1, J2, . . . that form a complex

· · · → J2 → J1 → A→ 0.

(In each case the image of Ji+1 is the connected component of the kernel of Ji →Ji−1.) Is it possible to make this construction in such a way that the sequenceterminates in 0?

Question 9.5.7 (Yau). Let A be an abelian variety. What can be said aboutthe minimum of the dimensions of all Jacobians J such that there is a surjectivemorphism J → A?

Remark 9.5.8. Brian Conrad has explained to the author that if A is an abelianvariety over an infinite field, then A can be embedded in a Jacobian J . This doesnot follow directly from Theorem 9.5.4 above, since if J →→ A∨, then the dual mapA→ J need not be injective.

9.6 Neron Models

The main references for Neron models are as follows:

1. [AEC2]: Silverman, Advanced Topics in the Arithmetic of Elliptic Curves.Chapter IV of this book contains an extremely well written and motivateddiscussion of Neron models of elliptic curves over Dedekind domains withperfect residue field. In particular, Silverman gives an almost complete con-struction of Neron models of elliptic curves. Silverman very clearly reallywants his reader to understand the construction. Highly recommended.

2. [BLR]: Bosch, Lutkebohmert, Raynaud, Neron Models. This is an excellentand accessible book that contains a complete construction of Neron modelsand some of their generalizations, a discussion of their functorial properties,and a sketch of the construction of Jacobians of families of curves. The goalof this book was to redo in scheme-theoretic language Neron original paper,which is written in a language that was ill-adapted to the subtleties of Neronmodels.

3. Artin, Neron Models, in Cornell-Silverman. This is the first-ever expositionof Neron’s original paper in the language of schemes.


9.6.1 What are Neron Models?

Suppose E is an elliptic curve over Q. If ∆ is the minimal discriminant of E,then E has good reduction at p for all p - ∆, in the sense that E extends to anabelian scheme E over Zp (i.e., a “smooth” and “proper” group scheme). One cannot ask for E to extend to an abelian scheme over Zp for all p | ∆. One can,however, ask whether there is a notion of “good” model for E at these bad primes.To quote [BLR, page 1], “It came as a surprise for arithmeticians and algebraicgeometers when A. Neron, relaxing the condition of properness and concentratingon the group structure and the smoothness, discovered in the years 1961–1963 thatsuch models exist in a canonical way.”

Before formally defining Neron models, we describe what it means for a mor-phism f : X → Y of schemes to be smooth. A morphism f : X → Y is finite typeif for every open affine U = Spec(R) ⊂ Y there is a finite covering of f−1(U) byopen affines Spec(S), such that each S is a finitely generated R-algebra.

Definition 9.6.1. A morphism f : X → Y is smooth at x ∈ X if it is of finitetype and there are open affine neighborhoods Spec(A) ⊂ X of x and Spec(R) ⊂ Yof f(x) such that

A ∼= R[t1, . . . , tn+r]/(f1, . . . , fn)

for elements f1, . . . , fn ∈ R[t1, . . . , tn+r] and all n × n minors of the Jacobianmatrix (∂fi/∂tj) generate the unit ideal of A. The morphism f is etale at x if, inaddition, r = 0. A morphism is smooth of relative dimension d if it is smooth at xfor every x ∈ X and r = d in the isomorphism above.

Smooth morphisms behave well. For example, if f and g are smooth and f gis defined, then f g is automatically smooth. Also, smooth morphisms are closedunder base extension: if f : X → Y is a smooth morphism over S, and S ′ is ascheme over S, then the induced mapX×SS′ → Y ×SS′ is smooth. (If you’ve neverseen products of schemes, it might be helpful to know that Spec(A)× Spec(B) =Spec(A⊗B). Read [Har77, §II.3] for more information about fiber products, whichprovide a geometric way to think about tensor products. Also, we often write XS′

as shorthand for X ×S S′.)We are now ready for the definition. Suppose R is a Dedekind domain with field

of fractions K (e.g., R = Z and K = Q).

Definition 9.6.2 (Neron model). Let A be an abelian variety over K. TheNeron model A of A is a smooth commutative group scheme over R such that forany smooth morphism S → R the natural map of abelian groups

HomR(S,A)→ HomK(S ×R K,A)

is a bijection. This is called the Neron mapping property: In more compact nota-tion, it says that there is an isomorphism A(S) ∼= A(SK).

Taking S = A in the definition we see that A is unique, up to a unique isomor-phism.

It is a deep theorem that Neron models exist. Fortunately, Bosch, Lutkebohmert,and Raynaud devoted much time to create a carefully written book [BLR90] thatexplains the construction in modern language. Also, in the case of elliptic curves,Silverman’s second book [Sil94] is extremely helpful.

9.6 Neron Models 97

The basic idea of the construction is to first observe that if we can constructa Neron model at each localization Rp at a nonzero prime ideal of R, then eachof these local models can be glued to obtain a global Neron model (this uses thatthere are only finitely many primes of bad reduction). Thus we may assume thatR is a discrete valuation ring.

The next step is to pass to the “strict henselization” R′ of R. A local ring R withmaximal ideal ℘ is henselian if “every simple root lifts uniquely”; more precisely,if whenever f(x) ∈ R[x] and α ∈ R is such that f(α) ≡ 0 (mod ℘) and f ′(α) 6≡ 0(mod ℘), there is a unique element α ∈ R such that α ≡ α (mod ℘) and f(α) = 0.The strict henselization of a discrete valuation ring R is an extension of R thatis henselian and for which the residue field of R′ is the separable closure of theresidue field of R (when the residue field is finite, the separable close is just thealgebraic closure). The strict henselization is not too much bigger than R, thoughit is typically not finitely generated over R. It is, however, much smaller thanthe completion of R (e.g., Zp is uncountable). The main geometric property ofa strictly henselian ring R with residue field k is that if X is a smooth schemeover R, then the reduction map X(R)→ X(k) is surjective.

Working over the strict henselization, we first resolve singularities. Then we usea generalization of the theorem that Weil used to construct Jacobians to pass froma birational group law to an actual group law. We thus obtain the Neron modelover the strict henselization of R. Finally, we use Grothendieck’s faithfully flatdescent to obtain a Neron model over R.

When A is the Jacobian of a curve X, there is an alternative approach thatinvolves the “minimal proper regular model” of X. For example, when A is anelliptic curve, it is the Jacobian of itself, and the Neron model can be constructed interms of the minimal proper regular model X of A as follows. In general, the modelX → R is not also smooth. Let X ′ be the smooth locus of X → R, which is obtainedby removing from each closed fiber XFp

=∑niCi all irreducible components with

multiplicity ni ≥ 2 and all singular points on each Ci, and all points where at leasttwo Ci intersect each other. Then the group structure on A extends to a groupstructure on X ′, and X ′ equipped with this group structure is the Neron modelof A.

Explicit determination of the possibilities for the minimal proper regular modelof an elliptic curve was carried out by Kodaira, then Neron, and finally in avery explicit form by Tate. Tate codified a way to find the model in what’scalled “Tate’s Algorithm” (see Antwerp IV, which is available on my web page:http://modular.fas.harvard.edu/scans/antwerp/, and look at Silverman, chap-ter IV, which also has important implementation advice).

9.6.2 The Birch and Swinnerton-Dyer Conjecture and Neron Models

Throughout this section, let A be an abelian variety over Q and let A be thecorresponding Neron model over Z. We work over Q for simplicity, but could workover any number field.

Let L(A, s) be the Hasse-Weil L-function of A (see Section [to be written]). Letr = ords=1 L(A, s) be the analytic rank of A. The Birch and Swinnerton-DyerConjecture asserts that A(Q) ≈ Zr ⊕A(Q)tor and

L(r)(A, 1)

r!=

(∏cp) · ΩA · RegA·#X(A)

#A(Q)tor ·#A∨(Q)tor.


We have not defined most of the quantities appearing in this formula. In thissection, we will define the Tamagawa numbers cp, the real volume ΩA, and theShafarevich-Tate group X(A) in terms of the Neron model A of A.

We first define the Tamagawa numbers cp, which are the orders groups of con-nected components. Let p be a prime and consider the closed fiber AFp

, whichis a smooth commutative group scheme over Fp. Then AFp

is a disjoint union ofone or more connected components. The connected component A0

Fpthat contains

the identity element is a subgroup of AFp(Intuition: the group law is continuous

and the continuous image of a connected set is connected, so the group structurerestricts to A0

Fp).

Definition 9.6.3 (Component Group). The component group of A at p is

ΦA,p = AFp/A0

Fp.

Fact: The component group ΦA,p is a finite flat group scheme over Fp, and forall but finitely many primes p, we have ΦA,p = 0.

Definition 9.6.4 (Tamagawa Numbers). The Tamagawa number of A at aprime p is

cp = #ΦA,p(Fp).

Next we define the real volume ΩA. Choose a basis

ω1, . . . , ωd ∈ H0(A,Ω1A/Z)

for the global differential 1-forms on A, where d = dimA. The wedge productw = ω1∧ω2∧· · ·∧ωd is a global d-form on A. Then w induces a differential d-formon the real Lie group A(R).

Definition 9.6.5 (Real Volume). The real volume of A is

ΩA =

∣∣∣∣∣

∫

A(R)

w

∣∣∣∣∣ ∈ R>0.

Finally, we give a definition of the Shafarevich-Tate group in terms of the Neronmodel. Let A0 be the scheme obtained from the Neron model A over A by removingfrom each closed fiber all nonidentity components. Then A0 is again a smoothcommutative group scheme, but it need not have the Neron mapping property.

Recall that an etale morphism is a morphism that is smooth of relative dimen-sion 0. A sheaf of abelian groups on the etale site Zet is a functor (satisfying certainaxioms) from the category of etale morphism X → Z to the category of abeliangroups. There are enough sheaves on Zet so that there is a cohomology theory forsuch sheaves, which is called etale cohomology. In particular if F is a sheaf on Zet,then for every integer q there is an abelian group Hq(Zet,F) associated to F thathas the standard properties of a cohomology functor.

The group schemes A0 and A both determine sheaves on the etale site, whichwe will also denote by A0 and A.

Definition 9.6.6 (Shafarevich-Tate Group). Suppose A(R) is connected thatthat A0 = A. Then the Shafarevich-Tate group of A is H1(Zet,A). More generally,

9.6 Neron Models 99

suppose only that A(R) is connected. Then the Shafarevich-Tate group is theimage of the natural map

f : H1(Zet,A0)→ H1(Zet,A).

Even more generally, if A(R) is not connected, then there is a natural map r :H1(Zet,A)→ H1(Gal(C/R), A(C)) and X(A) = im(f) ∩ ker(r).

Mazur proves in the appendix to [Maz72] that this definition is equivalent to theusual Galois cohomology definition. To do this, he considers the exact sequence0 → A0 → A → ΦA → 0, where ΦA is a sheaf version of ⊕pΦA,p. The maininput is Lang’s Theorem, which implies that over a local field, unramified Galoiscohomology is the same as the cohomology of the corresponding component group.

Conjecture 9.6.7 (Shafarevich-Tate). The group H1(Zet,A) is finite.

When A has rank 0, all component groups ΦA,p are trivial, A(R) is connected,and A(Q)tor and A∨(Q)tor are trivial, the Birch and Swinnerton-Dyer conjecturetakes the simple form

L(A, 1)

ΩA= #H1(Zet,A).

Later, when A is modular, we will (almost) interpret L(A, 1)/ΩA as the order ofa certain group that involves modular symbols. Thus the BSD conjecture assertsthat two groups have the same order; however, they are not isomorphic, since, e.g.,when dimA = 1 the modular symbols group is always cyclic, but the Shafarevich-Tate group is never cyclic (unless it is trivial).

9.6.3 Functorial Properties of Neron Models

The definition of Neron model is functorial, so one might expect the formation ofNeron models to have good functorial properties. Unfortunately, it doesn’t.

Proposition 9.6.8. Let A and B be abelian varieties. If A and B are the Neronmodels of A and B, respectively, then the Neron model of A×B is A× B.

Suppose R ⊂ R′ is a finite extension of discrete valuation rings with fields offractions K ⊂ K ′. Sometimes, given an abelian variety A over a field K, it iseasier to understand properties of the abelian variety, such as reduction, over K ′.For example, you might have extra information that implies that AK′ decomposesas a product of well-understood abelian varieties. It would thus be useful if theNeron model of AK′ were simply the base extension AR′ of the Neron model of Aover R. This is, however, frequently not the case.

Distinguishing various types of ramification will be useful in explaining howNeron models behave with respect to base change, so we now recall the notions oftame and wild ramification. If π generates the maximal ideal of R and v′ is thevaluation on R′, then the extension is unramified if v′(π) = 1. It is tamely ramifiedif v′(π) is not divisible by the residue characteristic of R, and it is wildly ramifiedif v′(π) is divisible by the residue characteristic of R. For example, the extensionQp(p

1/p) of Qp is wildly ramified.

Example 9.6.9. If R is the ring of integers of a p-adic field, then for every integer nthere is a unique unramified extension of R of degree n. See [Cp86, §I.7], whereFrohlich uses Hensel’s lemma to show that the unramified extensions of K =


Frac(R) are in bijection with the finite (separable) extensions of the residue classfield.

The Neron model does not behave well with respect to base change, except insome special cases. For example, suppose A is an abelian variety over the fieldof fractions K of a discrete valuation ring R. If K ′ is the field of fractions of afinite unramified extension R′ of R, then the Neron model of AK′ is AR′ , whereA is the Neron model of A over R. Thus the Neron model over an unramifiedextension is obtained by base extending the Neron model over the base. This isnot too surprising because in the construction of Neron model we first passed tothe strict henselization of R, which is a limit of unramified extensions.

Continuing with the above notation, if K ′ is tamely ramified over K, then ingeneral AR′ need not be the Neron model of AK′ . Assume that K ′ is Galois over K.In [Edi92], Bas Edixhoven describes the Neron model of AK in terms of AR′ . Todescribe his main theorem, we introduce the restriction of scalars of a scheme.

Definition 9.6.10 (Restriction of Scalars). Let S ′ → S be a morphism ofschemes and let X ′ be a scheme over S′. Consider the functor

R(T ) = HomS′(T ×S S′, X ′)

on the category of all schemes T over S. If this functor is representable, the rep-resenting object X = ResS′/S(X ′) is called the restriction of scalars of X ′ to S.

Edixhoven’s main theorem is that if G is the Galois group of K ′ over K andX = ResR′/R(AR′) is the restriction of scalars of AR′ down to R, then there is anatural map A → X whose image is the closed subscheme XG of fixed elements.

We finish this section with some cautious remarks about exactness properties ofNeron models. If 0 → A → B → C → 0 is an exact sequence of abelian varieties,then the functorial definition of Neron models produces a complex of Neron models

0→ A→ B → C → 0,

where A, B, and C are the Neron models of A, B, and C, respectively. This complexcan fail to be exact at every point. For an in-depth discussion of conditions whenwe have exactness, along with examples that violate exactness, see [BLR90, Ch. 7],which says: “we will see that, except for quite special cases, there will be a defectof exactness, the defect of right exactness being much more serious than the oneof left exactness.”

To give examples in which right exactness fails, it suffices to give an optimalquotient B → C such that for some p the induced map ΦB,p → ΦC,p on componentgroups is not surjective (recall that optimal means A = ker(B → C) is an abelianvariety). Such quotients, with B and C modular, arise naturally in the context ofRibet’s level optimization. For example, the elliptic curve E given by y2 + xy =x3 + x2 − 11x is the optimal new quotient of the Jacobian J0(33) of X0(33). Thecomponent group of E at 3 has order 6, since E has semistable reduction at 3(since 3 || 33) and ord3(j(E)) = −6. The image of the component group of J0(33)in the component group of E has order 2:

> OrderOfImageOfComponentGroupOfJ0N(ModularSymbols("33A"),3);

2

Note that the modular form associated to E is congruent modulo 3 to the formcorresponding to J0(11), which illustrates the connection with level optimization.

10Abelian Varieties Attached to ModularForms

In this chapter we describe how to decompose J1(N), up to isogeny, as a product ofabelian subvarieties Af corresponding to Galois conjugacy classes of cusp forms fof weight 2. This was first accomplished by Shimura (see [Shi94, Theorem 7.14]).We also discuss properties of the Galois representation attached to f .

In this chapter we will work almost exclusively with J1(N). However, everythinggoes through exactly as below with J1(N) replaced by J0(N) and S2(Γ1(N)) re-placed by S2(Γ0(N)). Since, J1(N) has dimension much larger than J0(N), so forcomputational investigations it is frequently better to work with J0(N).

See Brian Conrad’s appendix to [ribet-stein: Lectures on Serre’s Conjectures]for a much more extensive exposition of the construction discussed below, which isgeared toward preparing the reader for Deligne’s more general construction of Ga-lois representations associated to newforms of weight k ≥ 2 (for that, see Conrad’sbook ...).

10.1 Decomposition of the Hecke Algebra

Let N be a positive integer and let

T = Z[. . . , Tn, . . .] ⊂ End(J1(N))

be the algebra of all Hecke operators acting on J1(N). Recall from Section 7.4 thatthe anemic Hecke algebra is the subalgebra

T0 = Z[. . . , Tn, . . . : (n,N) = 1] ⊂ T

of T obtained by adjoining to Z only those Hecke operators Tn with n relativelyprime to N .

Remark 10.1.1. Viewed as Z-modules, T0 need not be saturated in T, i.e., T/T0

need not be torsion free. For example, if T is the Hecke algebra associated to

102 10. Abelian Varieties Attached to Modular Forms

S2(Γ1(24)) then T/T0∼= Z/2Z. Also, if T is the Hecke algebra associated to

S2(Γ0(54)), then T/T0∼= Z/3Z× Z.

If f =∑anq

n is a newform, then the field Kf = Q(a1, a2, . . .) has finite degreeover Q, since the an are the eigenvalues of a family of commuting operators withintegral characteristic polynomials. The Galois conjugates of f are the newformsσ(f) =

∑σ(an)q

n, for σ ∈ Gal(Q/Q). There are [Kf : Q] Galois conjugates of f .As in Section 7.4, we have a canonical decomposition

T0 ⊗Q ∼=∏

f

Kf , (10.1.1)

where f varies over a set of representatives for the Galois conjugacy classes ofnewforms in S2(Γ1(N)) of level dividing N . For each f , let

πf = (0, . . . , 0, 1, 0, . . . , 0) ∈∏

Kf

be projection onto the factor Kf of the product (10.1.1). Since T0 ⊂ T, and Thas no additive torsion, we have T0 ⊗ Q ⊂ T ⊗ Q, so these projectors πf liein TQ = T ⊗ Q. Since TQ is commutative and the πf are mutually orthogonalidempotents whose sum is (1, 1, . . . , 1), we see that TQ breaks up as a product ofalgebras

TQ∼=∏

f

Lf , t 7→∑

f

tπf .

10.1.1 The Dimension of Lf

Proposition 10.1.2. If f , Lf and Kf are as above, then dimKfLf is the number

of divisors of N/Nf where Nf is the level of the newform f .

Proof. Let Vf be the complex vector space spanned by all images of Galois con-jugates of f via all maps αd with d | N/Nf . It follows from [Atkin-Lehner-Li theory – multiplicity one] that the images via αd of the Galois conjugatesof f are linearly independent. (Details: More generally, if f and g are newformsof level M , then by Proposition 7.2.1, B(f) = αd(f) : d | N/Nf is a lin-early independent set and likewise for B(g). Suppose some nonzero element f ′

of the span of B(f) equals some element g′ of the span of B(g). Since Tp, forp - N , commutes with αd, we have Tp(f

′) = ap(f)f ′ and Tp(g′) = ap(g)g

′, so0 = Tp(0) = Tp(f

′ − g′) = ap(f)f ′ − ap(g)g′. Since f ′ = g′, this implies that

ap(f) = ap(g). Because a newform is determined by the eigenvalues of Tp forp - N , it follows that f = g.) Thus the C-dimension of Vf is the number of divisorsof N/Nf times dimQKf .

The factor Lf is isomorphic to the image of TQ ⊂ End(Sk(Γ1(N))) in End(Vf ).As in Section ??, there is a single element v ∈ Vf so that Vf = TC · v. Thus theimage of TQ in End(Vf ) has dimension dimC Vf , and the result follows.

Let’s examine a particular case of this proposition. Suppose p is a prime and f =∑anq

n is a newform of level Nf coprime to p, and let N = p ·Nf . We will showthat

Lf = Kf [U ]/(U2 − apU + p), (10.1.2)

10.2 Decomposition of J1(N) 103

hence dimKfLf = 2 which, as expected, is the number of divisors of N/Nf = p.

The first step is to view Lf as the space of operators generated by the Heckeoperators Tn acting on the span V of the images f(dz) = f(qd) for d | (N/Nf ) = p.If n 6= p, then Tn acts on V as the scalar an, and when n = p, the Hecke operator Tpacts on Sk(Γ1(p ·Nf )) as the operator also denoted Up. By Section 7.2, we know

that Up corresponds to the matrix(ap 1−p 0

)with respect to the basis f(q), f(qp)

of V . Thus Up satisfies the relation U 2p − apU + p. Since Up is not a scalar matrix,

this minimal polynomial of Up is quadratic, which proves (10.1.2).More generally, see [DDT94, Lem. 4.4] (Diamond-Darmon-Taylor) for an explicit

presentation of Lf as a quotient

Lf ∼= Kf [. . . , Up, . . .]/I

where I is an ideal and the Up correspond to the prime divisors of N/Nf .

10.2 Decomposition of J1(N)

Let f be a newform in S2(Γ1(N)) of level a divisor M of N , so f ∈ S2(Γ1(M))new

is a normalized eigenform for all the Hecke operators of level M . We associateto f an abelian subvariety Af of J1(N), of dimension [Lf : Q], as follows. Recallthat πf is the fth projector in T0 ⊗Q =

∏gKg. We can not define Af to be the

image of J1(N) under πf , since πf is only, a priori, an element of End(J1(N))⊗Q.Fortunately, there exists a positive integer n such that nπf ∈ End(J1(N)), and welet

Af = nπf (J1(N)).

This is independent of the choice of n, since the choices for n are all multiples ofthe “denominator” n0 of πf , and if A is any abelian variety and n is a positiveinteger, then nA = A.

The natural map∏f Af → J1(N), which is induced by summing the inclusion

maps, is an isogeny. Also Af is simple if f is of level N , and otherwise Af isisogenous to a power of A′

f ⊂ J1(Nf ). Thus we obtain an isogeny decompositionof J1(N) as a product of Q-simple abelian varieties.

Remark 10.2.1. The abelian varieties Af frequently decompose further over Q,i.e., they are not absolutely simple, and it is an interesting problem to determinean isogeny decomposition of J1(N)Q as a product of simple abelian varieties. It isstill not known precisely how to do this computationally for any particular N .

This decomposition can be viewed in another way over the complex numbers.As a complex torus, J1(N)(C) has the following model:

J1(N)(C) = Hom(S2(Γ1(N)),C)/H1(X1(N),Z).

The action of the Hecke algebra T on J1(N)(C) is compatible with its action onthe cotangent space S2(Γ1(N)). This construction presents J1(N)(C) naturallyas V/L with V a complex vector space and L a lattice in V . The anemic Heckealgebra T0 then decomposes V as a direct sum V =

⊕f Vf . The Hecke operators

act on Vf and L in a compatible way, so T0 decomposes L ⊗Q in a compatibleway. Thus Lf = Vf ∩ L is a lattice in Vf , so we may Af (C) view as the complextorus Vf/Lf .


Lemma 10.2.2. Let f ∈ S2(Γ1(N)) be a newform of level dividing N and Af =nπf (J1(N)) be the corresponding abelian subvariety of J1(N). Then the Heckealgebra T ⊂ End(J1(N)) leaves Af invariant.

Proof. The Hecke algebra T is commutative, so if t ∈ T, then

tAf = tnπf (J1(N)) = nπf (tJ1(N)) ⊂ nπf (J1(N)) = Af .

Remark 10.2.3. Viewing Af (C) as Vf/Lf is extremely useful computationally,since L can be computed using modular symbols, and Lf can be cut out usingthe Hecke operators. For example, if f and g are nonconjugate newforms of leveldividing N , we can explicitly compute the group structure of Af ∩Ag ⊂ J1(N) bydoing a computation with modular symbols in L. More precisely, we have

Af ∩Ag ∼= (L/(Lf + Lg))tor.

Note that Af depends on viewing f as an element of S2(Γ1(N)) for some N .Thus it would be more accurate to denote Af by Af,N , where N is any multiple ofthe level of f , and to reserve the notation Af for the case N = 1. Then dimAf,Nis dimAf times the number of divisors of N/Nf .

10.2.1 Aside: Intersections and Congruences

Suppose f and g are not Galois conjugate. Then the intersection Ψ = Af ∩ Agis finite, since Vf ∩ Vg = 0, and the integer #Ψ is of interest. This cardinalityis related to congruence between f and g, but the exact relation is unclear. Forexample, one might expect that p | #Ψ if and only if there is a prime ℘ of thecompositum Kf .Kg of residue characteristic p such that aq(f) ≡ aq(g) (mod ℘)for all q - N . If p | #Ψ, then such a prime ℘ exists (take ℘ to be induced bya maximal ideal in the support of the nonzero T-module Ψ[p]). The converse isfrequently true, but is sometimes false. For example, if N is the prime 431 and

f = q − q2 + q3 − q4 + q5 − q6 − 2q7 + · · ·g = q − q2 + 3q3 − q4 − 3q5 − 3q6 + 2q7 + · · · ,

then f ≡ g (mod 2), but Af ∩ Ag = 0. This example implies that “multiplicityone fails” for level 431 and p = 2, so the Hecke algebra associated to J0(431) isnot Gorenstein (see [Lloyd Kilford paper] for more details).

10.3 Galois Representations Attached to Af

It is important to emphasize the case when f is a newform of level N , since then Afis Q-simple and there is a compatible family of 2-dimensional `-adic representationsattached to f , which arise from torsion points on Af .

Proposition 10.1.2 implies that Lf = Kf . Fix such an f , let A = Af , letK = Kf ,and let

d = dimA = dimQK = [K : Q].

10.3 Galois Representations Attached to Af 105

Let ` be a prime and consider the Q`-adic Tate module Tate`(A) of A:

Tate`(A) = Q` ⊗ lim←−ν>0

A[`ν ].

Note that as a Q`-vector space Tate`(A) ∼= Q2d` , since A[n] ∼= (Z/nZ)2d, as groups.

There is a natural action of the ring K⊗QQ` on Tate`(A). By algebraic numbertheory

K ⊗Q Q` =∏

λ|`Kλ,

where λ runs through the primes of the ring OK of integers of K lying over ` andKλ denotes the completion of K with respect to the absolute value induced by λ.Thus Tate`(A) decomposes as a product

Tate`(A) =∏

λ|`Tateλ(A)

where Tateλ(A) is a Kλ vector space.

Lemma 10.3.1. Let the notation be as above. Then for all λ lying over `,

dimKλTateλ(A) = 2.

Proof. Write A = V/L, with V = Vf a complex vector space and L a lattice. ThenTateλ(A) ∼= L ⊗ Q` as Kλ-modules (not as Gal(Q/Q)-modules!), since A[`n] ∼=L/`nL, and lim←−n L/`

nL ∼= Z`⊗L. Also, L⊗Q is a vector space over K, which musthave dimension 2, since L ⊗Q has dimension 2d = 2dimA and K has degree d.Thus

Tateλ(A) ∼= L ⊗Kλ ≈ (K ⊕K)⊗K Kλ∼= Kλ ⊕Kλ

has dimension 2 over Kλ.

Now consider Tateλ(A), which is a Kλ-vector space of dimension 2. The Heckeoperators are defined over Q, so Gal(Q/Q) acts on Tate`(A) in a way compatiblewith the action of K ⊗Q Q`. We thus obtain a homomorphism

ρ` = ρf,` : Gal(Q/Q)→ AutK⊗Q`Tate`(A) ≈ GL2(K ⊗Q`) ∼=

∏

λ

GL2(Kλ).

Thus ρ` is the direct sum of `-adic Galois representations ρλ where

ρλ : Gal(Q/Q)→ EndKλ(Tateλ(A))

gives the action of Gal(Q/Q) on Tateλ(A).If p - `N , then ρλ is unramified at p (see [ST68, Thm. 1]). In this case it makes

sense to consider ρλ(ϕp), where ϕp ∈ Gal(Q/Q) is a Frobenius element at p. Thenρλ(ϕp) has a well-defined trace and determinant, or equivalently, a well-definedcharacteristic polynomial Φ(X) ∈ Kλ[X].

Theorem 10.3.2. Let f ∈ S2(Γ1(N), ε) be a newform of level N with Dirichletcharacter ε. Suppose p - `N , and let ϕp ∈ Gal(Q/Q) be a Frobenius element at p.Let Φ(X) be the characteristic polynomial of ρλ(ϕp). Then

Φ(X) = X2 − apX + p · ε(p),where ap is the pth coefficient of the modular form f (thus ap is the image of Tpin Ef and ε(p) is the image of 〈p〉).


Let ϕ = ϕp. By the Cayley-Hamilton theorem

ρλ(ϕ)2 − tr(ρλ(ϕ))ρλ(ϕ) + det(ρλ(ϕ)) = 0.

Using the Eichler-Shimura congruence relation (see ) we will show that tr(ρλ(ϕ)) =ap, but we defer the proof of this until ....

We will prove that det(ρλ(ϕ)) = p in the special case when ε = 1. This willfollow from the equality

det(ρλ) = χ`, (10.3.1)

where χ` is the `th cyclotomic character

χ` : Gal(Q/Q)→ Z∗` ⊂ K∗

λ,

which gives the action of Gal(Q/Q) on µ`∞ . We have χ`(ϕ) = p because ϕ inducesinduces pth powering map on µ`∞ .

It remains to establish (10.3.1). The simplest case is when A is an elliptic curve.In [Sil92, ], Silverman shows that det(ρ`) = χ` using the Weil pairing. We willconsider the Weil pairing in more generality in the next section, and use it toestablish (10.3.1).

10.3.1 The Weil Pairing

Let T`(A) = lim←−n≥1A[`n], so Tate`(A) = Q` ⊗ T`(A). The Weil pairing is a non-

degenerate perfect pairing

e` : T`(A)× T`(A∨)→ Z`(1).

(See e.g., [Mil86, §16] for a summary of some of its main properties.)

Remark 10.3.3. Identify Z/`nZ with µ`n by 1 7→ e−2πi/`n , and extend to a mapZ` → Z`(1). If J = Jac(X) is a Jacobian, then the Weil pairing on J is inducedby the canonical isomorphism

T`(J) ∼= H1(X,Z`) = H1(X,Z)⊗ Z`,

and the cup product pairing

H1(X,Z`)⊗Z`H1(X,Z`)

∪−−→ Z`.

For more details see the discussion on pages 210–211 of Conrad’s appendix to[RS01], and the references therein. In particular, note that H1(X,Z`) is isomorphicto H1(X,Z`), because H1(X,Z`) is self-dual because of the intersection pairing. Itis easy to see that H1(X,Z`) ∼= T`(J) since by Abel-Jacobi J ∼= T0(J)/H1(X,Z),where T0(J) is the tangent space at J at 0 (see Lemma 10.3.1).

Here Z`(1) ∼= lim←−µ`n is isomorphic to Z` as a ring, but has the action of

Gal(Q/Q) induced by the action of Gal(Q/Q) on lim←−µ`n . Given σ ∈ Gal(Q/Q),

there is an element χ`(σ) ∈ Z∗` such that σ(ζ) = ζχ`(σ), for every `nth root of

unity ζ. If we view Z`(1) as just Z` with an action of Gal(Q/Q), then the actionof σ ∈ Gal(Q/Q) on Z`(1) is left multiplication by χ`(σ) ∈ Z∗

` .

10.3 Galois Representations Attached to Af 107

Definition 10.3.4 (Cyclotomic Character). The homomorphism

χ` : Gal(Q/Q)→ Z∗`

is called the `-adic cyclotomic character.

If ϕ : A→ A∨ is a polarization (so it is an isogeny defined by translation of anample invertible sheaf), we define a pairing

eϕ` : T`(A)× T`(A)→ Z`(1) (10.3.2)

by eϕ` (a, b) = e`(a, ϕ(b)). The pairing (10.3.2) is a skew-symmetric, nondegenerate,bilinear pairing that is Gal(Q/Q)-equivariant, in the sense that if σ ∈ Gal(Q/Q),then

eϕ` (σ(a), σ(b)) = σ · eϕ` (a, b) = χ`(σ)eϕ` (a, b).

We now apply the Weil pairing in the special case A = Af ⊂ J1(N). Abelianvarieties attached to modular forms are equipped with a canonical polarizationcalled the modular polarization. The canonical principal polarization of J1(N) isan isomorphism J1(N)

∼−→ J1(N)∨, so we obtain the modular polarization ϕ =ϕA : A→ A∨ of A, as illustrated in the following diagram:

J1(N)autoduality∼=

// J1(N)∨

²²A

OO

polarizationϕA

// A∨

Consider (10.3.2) with ϕ = ϕA the modular polarization. Tensoring over Qand restricting to Tateλ(A), we obtain a nondegenerate skew-symmetric bilinearpairing

e : Tateλ(A)× Tateλ(A)→ Q`(1). (10.3.3)

The nondegeneracy follows from the nondegeneracy of eϕ` and the observation that

eϕ` (Tateλ(A),Tateλ′(A)) = 0

when λ 6= λ′. This uses the Galois equivariance of eφ` carries over to Galois equiv-ariance of e, in the following sense. If σ ∈ Gal(Q/Q) and x, y ∈ Tateλ(A), then

e(σx, σy) = σe(x, y) = χ`(σ)e(x, y).

Note that σ acts on Q`(1) as multiplication by χ`(σ).

10.3.2 The Determinant

There are two proofs of the theorem, a fancy proof and a concrete proof. We firstpresent the fancy proof. The pairing e of (10.3.3) is a skew-symmetric and bilinearform so it determines a Gal(Q/Q)-equivarient homomorphism

2∧

Kλ

Tateλ(A)→ Q`(1). (10.3.4)


It is not a priori true that we can take the wedge product over Kλ instead ofQ`, but we can because e(tx, y) = e(x, ty) for any t ∈ Kλ. This is where we usethat A is attached to a newform with trivial character, since when the characteris nontrivial, the relation between e(Tpx, y) and e(x, Tpy) will involve 〈p〉. Let

D =∧2

Tateλ(A) and note that dimKλD = 1, since Tateλ(A) has dimension 2

over Kλ.There is a canonical isomorphism

HomQ`(D,Q`(1)) ∼= HomKλ

(D,Kλ(1)),

and the map of (10.3.4) maps to an isomorphismD ∼= Kλ(1) of Gal(Q/Q)-modules.Since the representation of Gal(Q/Q) on D is the determinant, and the represen-tation on Kλ(1) is the cyclotomic character χ`, it follows that det ρλ = χ`.

Next we consider a concrete proof. If σ ∈ Gal(Q/Q), then we must show thatdet(σ) = χ`(σ). Choose a basis x, y ∈ Tateλ(A) of Tateλ(A) as a 2 dimensionalKλ vector space. We have σ(x) = ax+ cy and σ(y) = bx+ dy, for a, b, c, d ∈ Kλ.Then

χ`(σ)e(x, y) = 〈σx, σy)= e(ax+ cy, bx+ dy)

= e(ax, bx) + e(ax, dy) + e(cy, bx) + e(cy, dy)

= e(ax, dy) + e(cy, bx)

= e(adx, y)− e(bcx, y)= e((ad− bc)x, y)= (ad− bc)e(x, y)

To see that e(ax, bx) = 0, note that

e(ax, bx) = e(abx, x) = −e(x, abx) = −e(ax, bx).

Finally, since e is nondegenerate, there exists x, y such that e(x, y) 6= 0, so χ`(σ) =ad− bc = det(σ).

10.4 Remarks About the Modular Polarization

Let A and ϕ be as in Section 10.3.1. The degree deg(ϕ) of the modular polarizationof A is an interesting arithmetic invariant of A. If B ⊂ J1(N) is the sum of allmodular abelian varieties Ag attached to newforms g ∈ S2(Γ1(N)), with g not aGalois conjugate of f and of level dividing N , then ker(ϕ) ∼= A ∩B, as illustrated

10.4 Remarks About the Modular Polarization 109

in the following diagram:

ker(ϕB)

$$JJJ

JJJJ

JJJ

∼=²²

ker(ϕAi)∼= //

%%LLLL

LLLL

LLL

A ∩B //

²²

B

²² ""FFF

FFFF

FF

A //

ϕ

%%KKK

KKKK

KKKK

J1(N)

²²

// B∨

A∨

Note that ker(ϕB) is also isomorphic to A ∩B, as indicated in the diagram.In connection with Section ??, the quantity ker(ϕA) = A∩B is closely related to

congruences between f and eigenforms orthogonal to the Galois conjugates of f .When A has dimension 1, we may alternatively view A as a quotient of X1(N)

via the mapX1(N)→ J1(N)→ A∨ ∼= A.

Then ϕA : A → A is pullback of divisors to X1(N) followed by push forward,which is multiplication by the degree. Thus ϕA = [n], where n is the degree of themorphism X1(N)→ A of algebraic curves. The modular degree is

deg(X1(N)→ A) =√

deg(ϕA).

More generally, if A has dimension greater than 1, then deg(ϕA) has order a perfectsquare (for references, see [Mil86, Thm. 13.3]), and we define the modular degreeto be

√deg(ϕA).

Let f be a newform of level N . In the spirit of Section 10.2.1 we use congruencesto define a number related to the modular degree, called the congruence number.For a subspace V ⊂ S2(Γ1(N)), let V (Z) = V ∩Z[[q]] be the elements with integralq-expansion at ∞ and V ⊥ denotes the orthogonal complement of V with respectto the Petersson inner product. The congruence number of f is

rf = #S2(Γ1(N))(Z)

Vf (Z) + V ⊥f (Z)

,

where Vf is the complex vector space spanned by the Galois conjugates of f . Wethus have two positive associated to f , the congruence number rf and the modulardegree mf of of Af .

Theorem 10.4.1. mf | rfRibet mentions this in the case of elliptic curves in [ZAGIER, 1985] [Zag85a],

but the statement is given incorrectly in that paper (the paper says that rf | mf ,which is wrong). The proof for dimension greater than one is in [AGASHE-STEIN,Manin constant...]. Ribet also subsequently proved that if p2 - N , then ordp(mf ) =ordp(rf ).

We can make the same definitions with J1(N) replaced by J0(N), so if f ∈S2(Γ0(N)) is a newform, Af ⊂ J0(N), and the congruence number measures con-gruences between f and other forms in S2(Γ0(N)). In [FM99, Ques. 4.4], they ask


whether it is always the case that mf = rf when Af is an elliptic curve, and mf

and rf are defined relative to Γ0(N). I implemented an algorithm in MAGMA tocompute rf , and found the first few counterexamples, which occur when

N = 54, 64, 72, 80, 88, 92, 96, 99, 108, 120, 124, 126, 128, 135, 144.

For example, the elliptic curve A labeled 54B1 in [Cre97] has rA = 6 and mA = 2.To see directly that 3 | rA, observe that if f is the newform corresponding to Eand g is the newform corresponding to X0(27), then g(q)+ g(q2) is congruent to fmodulo 3. This is consistent with Ribet’s theorem that if p | rA/mA then p2 | N .There seems to be no absolute bound on the p that occur.

It would be interesting to determine the answer to the analogue of the questionof Frey-Mueller for Γ1(N). For example, if A ⊂ J1(54) is the curve isogeneous to54B1, then mA = 18 is divisible by 3. However, I do not know rA in this case,because I haven’t written a program to compute it for Γ1(N).

11Modularity of Abelian Varieties

11.1 Modularity Over Q

Definition 11.1.1 (Modular Abelian Variety). Let A be an abelian varietyover Q. Then A is modular if there exists a positive integer N and a surjectivemap J1(N)→ A defined over Q.

The following theorem is the culmination of a huge amount of work, whichstarted with Wiles’s successful attack [Wil95] on Fermat’s Last Theorem, andfinished with [BCDT01].

Theorem 11.1.2 (Breuil, Conrad, Diamond, Taylor, Wiles). Let E be anelliptic curve over Q. Then E is modular.

We will say nothing about the proof here.If A is an abelian variety over Q, let EndQ(A) denote the ring of endomorphisms

of A that are defined over Q.

Definition 11.1.3 (GL2-type). An abelian variety A over Q is of GL2-type ifthe endomorphism algebra Q⊗ EndQ(A) contains a number field of degree equalto the dimension of A.

For example, every elliptic curve E over Q is trivially of GL2-type, since Q ⊂Q⊗ EndQ(E).

Proposition 11.1.4. If A is an abelian variety over Q, and K ⊂ Q⊗ EndQ(A)is a field, then [K : Q] divides dimA.

Proof. As discussed in [Rib92, §2],K acts faithfully on the tangent space Tan0(A/Q)over Q to A at 0, which is a Q vector space of dimension dim(A). Thus Tan0(A/Q)is a vector space over K, hence has Q-dimension a multiple of [K : Q].

Proposition 11.1.4 implies, in particular, that if E is an elliptic curve over Q,then EndQ(E) = Q. Recall that E has CM or is a complex multiplication elliptic

112 11. Modularity of Abelian Varieties

curve if EndQ(E) 6= Z). Proposition 11.1.4 implies that if E is a CM elliptic curve,the extra endomorphisms are never defined over Q.

Proposition 11.1.5. Suppose A = Af ⊂ J1(N) is an abelian variety attached toa newform of level N . Then A is of GL2-type.

Proof. The endomorphism ring of Af contains Of = Z[. . . , an(f), . . .], hence thefield Kf = Q(. . . , an(f), . . .) is contained in Q⊗ EndQ(A). Since Af = nπJ1(N),where π is a projector onto the factor Kf of the anemic Hecke algebra T0 ⊗Z Q,we have dimAf = [Kf : Q]. (One way to see this is to recall that the tangentspace T = Hom(S2(Γ1(N)),C) to J1(N) at 0 is free of rank 1 over T0 ⊗Z C.)

Conjecture 11.1.6 (Ribet). Every abelian variety over Q of GL2-type is mod-ular.

Supposeρ : Gal(Q/Q)→ GL2(Fp)

is an odd irreducible continuous Galois representation, where odd means that

det(ρ(c)) = −1,

where c is complex conjugation. We say that ρ is modular if there is a newformf ∈ Sk(Γ1(N)), and a prime ideal ℘ ⊂ Of such that for all ` - Np, we have

Tr(ρ(Frob`)) ≡ a` (mod ℘),

Det(ρ(Frob`)) ≡ `k−1 · ε(`) (mod ℘).

Here χp is the p-adic cyclotomic character, and ε is the (Nebentypus) character ofthe newform f .

Conjecture 11.1.7 (Serre). Every odd irreducible continuous representation

ρ : Gal(Q/Q)→ GL2(Fp)

is modular. Moreover, there is a formula for the “optimal” weight k(ρ) and levelN(ρ) of a newform that gives rise to ρ.

In [Ser87], Serre describes the formula for the weight and level. Also, it is nowknown due to work of Ribet, Edixhoven, Coleman, Voloch, Gross, and others thatif ρ is modular, then ρ arises from a form of the conjectured weight and level,except in some cases when p = 2. (For more details see the survey paper [RS01].)However, the full Conjecture 11.1.7 is known in very few cases.

Remark 11.1.8. There is interesting recent work of Richard Taylor which connectsConjecture 11.1.7 with the open question of whether every variety of a certain typehas a point over a solvable extension of Q. The question of the existence of solvablepoints (“solvability of varieties in radicals”) seems very difficult. For example, wedon’t even know the answer for genus one curves, or have a good reason to makea conjecture either way (as far as I know). There’s a book of Mike Fried thatdiscusses this solvability question.

Serre’s conjecture is very strong. For example, it would imply modularity of allabelian varieties over Q that could possibly be modular, and the proof of thisimplication does not rely on Theorem 11.1.2.

11.1 Modularity Over Q 113

Theorem 11.1.9 (Ribet). Serre’s conjectures on modularity of all odd irreduciblemod p Galois representations implies Conjecture 11.1.6.

To give the reader a sense of the connection between Serre’s conjecture andmodularity, we sketch some of the key ideas of the proof of Theorem 11.1.9; formore details the reader may consult Sections 1–4 of [Rib92].

Without loss, we may assume that A is Q-simple. As explained in the not trivial[Rib92, Thm. 2.1], this hypothesis implies that

K = Q⊗Z EndQ(A)

is a number field of degree dim(A). The Tate modules

Tate`(A) = Q` ⊗ lim←−n≥1

A[`n]

are free of rank two over K ⊗Q`, so the action of Gal(Q/Q) on Tate`(A) definesa representation

ρA,` : Gal(Q/Q)→ GL2(K ⊗Q`).

Remarks 11.1.10. That these representations take values in GL2 is why such A aresaid to be “of GL2-type”. Also, note that the above applies to A = Af ⊂ J1(N),and the `-adic representations attached to f are just the factors of ρA,` comingfrom the fact that K ⊗Q`

∼=∏λ|`Kλ.

The deepest input to Ribet’s proof is Faltings’s isogeny theorem, which Faltingsproved in order to prove Mordell’s conjecture (there are only a finite number ofL-rational points on any curve over L of genus at least 2).

If B is an abelian variety over Q, let

L(B, s) =∏

all primes p

1

det (1− p−s · Frobp |Tate`(A))=∏

p

Lp(B, s),

where ` is a prime of good reduction (it makes no difference which one).

Theorem 11.1.11 (Faltings). Let A and B be abelian varieties. Then A is isoge-nous to B if and only if Lp(A, s) = Lp(B, s) for almost all p.

Using an analysis of Galois representations and properties of conductors andapplying results of Faltings, Ribet finds an infinite set Λ of primes of K such thatall ρA,λ are irredudible and there only finitely many Serre invariants N(ρA,λ) andk(ρA,λ). For each of these λ, by Conjecture 11.1.7 there is a newform fλ of levelN(ρA,λ)) and weight k(ρA,λ) that gives rise to the mod ` representation ρA,λ.Since Λ is infinite, but there are only finitely many Serre invariants N(ρA,λ)),k(ρA,λ), there must be a single newform f and an infinite subset Λ′ of Λ so thatfor every λ ∈ Λ′ the newform f gives rise to ρA,λ.

Let B = Af ⊂ J1(N) be the abelian variety attached to f . Fix any prime p ofgood reduction. There are infinitely many primes λ ∈ Λ′ such that ρA,λ ∼= ρB,λ for

some λ, and for these λ,

det(1− p−s · Frobp |A[λ]

)= det

(1− p−s · Frobp |B[λ]

).

This means that the degree two polynomials in p−s (over the appropriate fields,e.g., K ⊗Q` for A)

det(1− p−s · Frobp |Tate`(A)

)

114 11. Modularity of Abelian Varieties

anddet(1− p−s · Frobp |Tate`(B)

)

are congruent modulo infinitely many primes. Therefore they are equal. By Theo-rem 11.1.11, it follows that A is isogenous to B = Af , so A is modular.

11.2 Modularity of Elliptic Curves over Q

Definition 11.2.1 (Modular Elliptic Curve). An elliptic curve E over Q ismodular if there is a surjective morphism X1(N)→ E for some N .

Definition 11.2.2 (Q-curve). An elliptic curve E over Q-bar is a Q-curve if forevery σ ∈ Gal(Q/Q) there is an isogeny Eσ → E (over Q).

Theorem 11.2.3 (Ribet). Let E be an elliptic curve over Q. If E is modular,then E is a Q-curve, or E has CM.

This theorem is proved in [Rib92, §5].Conjecture 11.2.4 (Ribet). Let E be an elliptic curve over Q. If E is a Q-curve,then E is modular.

In [Rib92, §6], Ribet proves that Conjecture 11.1.7 implies Conjecture 11.2.4. Hedoes this by showing that if a Q-curve E does not have CM then there is a Q-simpleabelian variety A over Q of GL2-type such that E is a simple factor of A over Q.This is accomplished finding a model for E over a Galois extension K of Q, re-stricting scalars down to Q to obtain an abelian variety B = ResK/Q(E), and using

Galois cohomology computations (mainly in H2’s) to find the required A of GL2-type inside B. Then Theorem 11.1.9 and our assumption that Conjecture 11.1.7 istrue together immediately imply that A is modular.

Ellenberg and Skinner [ES00] have recently used methods similar to those usedby Wiles to prove strong theorems toward Conjecture 11.2.4. See also Ellenberg’ssurvey [Ell02], which discusses earlier modularity results of Hasegawa, Hashimoto,Hida, Momose, and Shimura, and gives an example to show that there are infinitelymany Q-curves whose modularity is not known.

Theorem 11.2.5 (Ellenberg, Skinner). Let E be a Q-curve over a numberfield K with semistable reduction at all primes of K lying over 3, and supposethat K is unramified at 3. Then E is modular.

12L-functions

12.1 L-functions Attached to Modular Forms

Let f =∑n≥1 anq

n ∈ Sk(Γ1(N)) be a cusp form.

Definition 12.1.1 (L-series). The L-series of f is

L(f, s) =∑

n≥1

anns.

Definition 12.1.2 (Λ-function). The completed Λ function of f is

Λ(f, s) = N s/2(2π)−sΓ(s)L(f, s),

where

Γ(s) =

∫ ∞

0

e−ttsdt

t

is the Γ function (so Γ(n) = (n− 1)! for positive integers n).

We can view Λ(f, s) as a (Mellin) transform of f , in the following sense:

Proposition 12.1.3. We have

Λ(f, s) = N s/2

∫ ∞

0

f(iy)ysdy

y,

and this integral converges for Re(s) > k2 + 1.

116 12. L-functions

Proof. We have

∫ ∞

0

f(iy)ysdy

y=

∫ ∞

0

∞∑

n=1

ane−2πnyys

dy

y

=∞∑

n=1

an

∫ ∞

0

e−t(2πn)−stsdt

t(t = 2πny)

= (2π)−sΓ(s)

∞∑

n=1

anns.

To go from the first line to the second line, we reverse the summation and in-tegration and perform the change of variables t = 2πny. (We omit discussion ofconvergence.)

12.1.1 Analytic Continuation and Functional Equation

We define the Atkin-Lehner operator WN on Sk(Γ1(N)) as follows. If wN =(0 −1N 0

), then [w2

N ]k acts as (−N)k−2, so if

WN (f) = N1− k2 · f |[wN ]k,

then W 2N = (−1)k. (Note that WN is an involution when k is even.) It is easy

to check directly that if γ ∈ Γ1(N), then wNγw−1N ∈ Γ1(N), so WN preserves

Sk(Γ1(N)). Note that in general WN does not commute with the Hecke operatorsTp, for p | N .

The following theorem is mainly due to Hecke (and maybe other people, at leastin this generality). For a very general version of this theorem, see [Li75].

Theorem 12.1.4. Suppose f ∈ Sk(Γ1(N), χ) is a cusp form with character χ.Then Λ(f, s) extends to an entire (holomorphic on all of C) function which satisfiesthe functional equation

Λ(f, s) = ikΛ(WN (f), k − s).SinceNs/2(2π)−sΓ(s) is everywhere nonzero, Theorem 12.1.4 implies that L(f, s)

also extends to an entire function.It follows from Definition 12.1.2 that Λ(cf, s) = cΛ(f, s) for any c ∈ C. Thus

if f is a WN -eigenform, so that WN (f) = wf for some w ∈ C, then the functionalequation becomes

Λ(f, s) = ikwΛ(f, k − s).If k = 2, then WN is an involution, so w = ±1, and the sign in the functionalequation is ε(f) = ikw = −w, which is the negative of the sign of the Atkin-Lehner involution WN on f . It is straightforward to show that ε(f) = 1 if andonly if ords=1 L(f, s) is even. Parity observations such as this are extremely usefulwhen trying to understand the Birch and Swinnerton-Dyer conjecture.

Sketch of proof of Theorem 12.1.4 when N = 1. We follow [Kna92, §VIII.5] closely.Note that since w1 =

(0 1−1 0

)∈ SL2(Z), the condition W1(f) = f is satisfied for

any f ∈ Sk(1). This translates into the equality

f

(−1

z

)= zkf(z). (12.1.1)

12.1 L-functions Attached to Modular Forms 117

Write z = x + iy with x and y real. Then (12.1.1) along the positive imaginaryaxis (so z = iy with y positive real) is

f

(i

y

)= ikykf(iy). (12.1.2)

From Proposition 12.1.3 we have

Λ(f, s) =

∫ ∞

0

f(iy)ys−1dy, (12.1.3)

and this integral converges for Re(s) > k2 + 1.

Again using growth estimates, one shows that

∫ ∞

1

f(iy)ys−1dy

converges for all s ∈ C, and defines an entire function. Breaking the path in (12.1.3)at 1, we have for Re(s) > k

2 + 1 that

Λ(f, s) =

∫ 1

0

f(iy)ys−1dy +

∫ ∞

1

f(iy)ys−1dy.

Apply the change of variables t = 1/y to the first term and use (12.1.2) to get

∫ 1

0

f(iy)ys−1dy =

∫ 1

∞−f(i/t)t1−s

1

t2dt

=

∫ ∞

1

f(i/t)t−1−sdt

=

∫ ∞

1

iktkf(it)t−1−sdt

= ik∫ ∞

1

f(it)tk−1−sdt.

Thus

Λ(f, s) = ik∫ ∞

1

f(it)tk−s−1dt+

∫ ∞

1

f(iy)ys−1dy.

The first term is just a translation of the second, so the first term extends to anentire function as well. Thus Λ(f, s) extends to an entire function.

The proof of the general case for Γ0(N) is almost the same, except the path isbroken at 1/

√N , since i/

√N is a fixed point for wN .

12.1.2 A Conjecture About Nonvanishing of L(f, k/2)

Suppose f ∈ Sk(1) is an eigenform. If k ≡ 2 (mod 4), then L(f, k/2) = 0 forreasons related to the discussion after the statement of Theorem 12.1.4. On theother hand, if k ≡ 0 (mod 4), then ords=k/2 L(f, k/2) is even, so L(f, k/2) may ormay not vanish.

Conjecture 12.1.5. Suppose k ≡ 0 (mod 4). Then L(f, k/2) 6= 0.

118 12. L-functions

According to [CF99], Conjecture 12.1.5 is true for weight k if there is some nsuch that the characteristic polynomial of Tn on Sk(1) is irreducible. Thus Maeda’sconjecture implies Conjecture 12.1.5. Put another way, if you find an f of level 1and weight k ≡ 0 (mod 4) such that L(f, k/2) = 0, then Maeda’s conjecture isfalse for weight k.

Oddly enough, I personally find Conjecture 12.1.5 less convincing that Maeda’sconjecture, despite it being a weaker conjecture.

12.1.3 Euler Products

Euler products make very clear how L-functions of eigenforms encode deep arith-metic information about representations of Gal(Q/Q). Given a “compatible fam-ily” of `-adic representations ρ of Gal(Q/Q), one can define an Euler productL(ρ, s), but in general it is very hard to say anything about the analytic propertiesof L(ρ, s). However, as we saw above, when ρ is attached to a modular form, weknow that L(ρ, s) is entire.

Theorem 12.1.6. Let f =∑anq

n be a newform in Sk(Γ1(N), ε), and let L(f, s) =∑n≥1 ann

−s be the associated Dirichlet series. Then L(f, s) has an Euler product

L(f, s) =∏

p|N

1

1− app−s·∏

p-N

1

1− app−s + ε(p)pk−1p−2s.

Note that it is not really necessary to separate out the factors with p | N as wehave done, since ε(p) = 0 whenever p | N . Also, note that the denominators are ofthe form F (p−s), where

F (X) = 1− apX + ε(p)pk−1X2

is the reverse of the characteristic polynomial of Frobp acting on any of the `-adicrepresentations attached to f , with p 6= `.

Recall that if p is a prime, then for every r ≥ 2 the Hecke operators satisfy therelationship

Tpr = Tpr−1Tp − pk−1ε(p)Tpr−2 . (12.1.4)

Lemma 12.1.7. For every prime p we have the formal equality

∑

r≥0

TprXr =1

1− TpX + ε(p)pk−1X2. (12.1.5)

Proof. Multiply both sides of (12.1.5) by 1 − TpX + ε(p)pk−1X2 to obtain theequation

∑

r≥0

TprXr −∑

r≥0

(TprTp)Xr+1 +

∑

r≥0

(ε(p)pk−1Tpr )Xr+2 = 1.

This equation is true if and only if the lemma is true. Equality follows by checkingthe first few terms and shifting the index down by 1 for the second sum and downby 2 for the third sum, then using (12.1.4).

12.1 L-functions Attached to Modular Forms 119

0 1 2 3-1

0

1

L-series

x

y

E0

E1

E2

E3

E0 = [0, 0, 0, 0, 1], E1 = [0, 0, 1,−1, 0], E2 = [0, 1, 1,−2, 0], E3 = [0, 0, 1,−7, 6]FIGURE 12.1.1. Graph of L(E, s) for s real, for curves of ranks 0 to 3.

Note that ε(p) = 0 when p | N , so when p | N∑

r≥0

TprXr =1

1− TpX.

Since the eigenvalues an of f also satisfy (12.1.4), we obtain each factor of theEuler product of Theorem 12.1.6 by substituting the an for the Tn and p−s for Xinto (12.1.4). For (n,m) = 1, we have anm = anam, so

∑

n≥1

anns

=∏

p

∑

r≥0

apr

prs

,

which gives the full Euler product for L(f, s) =∑ann

−s.

12.1.4 Visualizing L-function

A. Shwayder did his Harvard junior project with me on visualizing L-functions ofelliptic curves (or equivalently, of newforms f =

∑anq

n ∈ S2(Γ0(N)) with an ∈ Zfor all n. The graphs in Figures 12.1.1–?? of L(E, s), for s real, and |L(E, s)|, for scomplex, are from his paper.

120 12. L-functions

13The Birch and Swinnerton-DyerConjecture

This chapter is about the conjecture of Birch and Swinnerton-Dyer on the arith-metic of abelian varieties. We focus primarily on abelian varieties attached tomodular forms.

In the 1960s, Sir Peter Swinnerton-Dyer worked with the EDSAC computer labat Cambridge University, and developed an operating system that ran on thatcomputer (so he told me once). He and Bryan Birch programmed EDSAC tocompute various quantities associated to elliptic curves. They then formulated theconjectures in this chapter in the case of dimension 1 (see [Bir65, Bir71, SD67]).Tate formulated the conjectures in a functorial way for abelian varieties of arbitrarydimension over global fields in [Tat66], and proved that if the conjecture is true foran abelian variety A, then it is also true for each abelian variety isogenous to A.

Suitably interpreted, the conjectures may by viewed as generalizing the ana-lytic class number formula, and Bloch and Kato generalized the conjectures toGrothendieck motives in [BK90].

13.1 The Rank Conjecture

Let A be an abelian variety over a number field K.

Definition 13.1.1 (Mordell-Weil Group). The Mordell-Weil group of A is theabelian group AK) of all K-rational points on A.

Theorem 13.1.2 (Mordell-Weil). The Mordell-Weil group A(K) of A is finitelygenerated.

The proof is nontrivial and combines two ideas. First, one proves the “weakMordell-Weil theorem”: for any integer m the quotient A(K)/mA(K) is finite.This is proved by combining Galois cohomology techniques with standard finitenesstheorems from algebraic number theory. The second idea is to introduce the Neron-

122 13. The Birch and Swinnerton-Dyer Conjecture

Tate canonical height h : A(K) → R≥0 and use properties of h to deduce, fromfiniteness of A(K)/mA(K), that A(K) itself is finitely generated.

Definition 13.1.3 (Rank). By the structure theorem A(K) ∼= Zr⊕Gtor, where ris a nonnegative integer and Gtor is the torsion subgroup of G. The rank of A is r.

Let f ∈ S2(Γ1(N)) be a newform of level N , and let A = Af ⊂ J1(N) bethe corresponding abelian variety. Let f1, . . . , fd denote the Gal(Q/Q)-conjugatesof f , so if f =

∑anq

n, then fi =∑σ(an)q

n, for some σ ∈ Gal(Q/Q).

Definition 13.1.4 (L-function of A). We define the L-function of A = Af (orany abelian variety isogenous to A) to be

L(A, s) =d∏

i=1

L(fi, s).

By Theorem 12.1.4, each L(fi, s) is an entire function on C, so L(A, s) is entire.In Section 13.4 we will discuss an intrinsic way to define L(A, s) that does notrequire that A be attached to a modular form. However, in general we do notknow that L(A, s) is entire.

Conjecture 13.1.5 (Birch and Swinnerton-Dyer). The rank of A(Q) is equalto ords=1 L(A, s).

One motivation for Conjecture 13.1.5 is the following formal observation. Assumefor simplicity of notation that dimA = 1. By Theorem 12.1.6, the L-functionL(A, s) = L(f, s) has an Euler product representation

L(A, s) =∏

p|N

1

1− app−s·∏

p-N

1

1− app−s + p · p−2s,

which is valid for Re(s) sufficiently large. (Note that ε = 1, since A is a modular el-liptic curve, hence a quotient ofX0(N).) There is no loss in considering the productL∗(A, s) over only the good primes p - N , since ords=1 L(A, s) = ords=1 L

∗(A, s)(because

∏p|N

11−app−s is nonzero at s = 1). We then have formally that

L∗(A, 1) =∏

p-N

1

1− app−1 + p−1

=∏

p-N

p

p− ap + 1

=∏

p-N

p

#A(Fp)

The intuition is that if the rank of A is large, i.e., A(Q) is large, then each groupA(Fp) will also be large since it has many points coming from reducing the ele-ments of A(Q) modulo p. It seems likely that if the groups #A(Fp) are unusuallylarge, then L∗(A, 1) = 0, and computational evidence suggests the more preciseConjecture 13.1.5.

Example 13.1.6. Let A0 be the elliptic curve y2 + y = x3 − x2, which has rank 0and conductor 11, let A1 be the elliptic curve y2+y = x3−x, which has rank 1 and

13.1 The Rank Conjecture 123

conductor 37, let A2 be the elliptic curve y2 + y = x3 + x2 − 2x, which has rank 2and conductor 389, and finally let A3 be the elliptic curve y2 + y = x3 − 7x + 6,which has rank 3 and conductor 5077. By an exhaustive search, these are known tobe the smallest-conductor elliptic curves of each rank. Conjecture 13.1.5 is knownto be true for them, the most difficult being A3, which relies on the results of[GZ86].

The following diagram illustrates |#Ai(Fp)| for p < 100, for each of these curves.The height of the red line (first) above the prime p is |#A0(Fp)|, the green line(second) gives the value for A1, the blue line (third) for A2, and the black line(fourth) for A3. The intuition described above suggests that the clumps shouldlook like triangles, with the first line shorter than the second, the second shorterthan the third, and the third shorter than the fourth—however, this is visibly notthe case. The large Mordell-Weil group over Q does not increase the size of everyE(Fp) as much as we might at first suspect. Nonetheless, the first line is no longerthan the last line for every p except p = 41, 79, 83, 97.

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97

Remark 13.1.7. Suppose that L(A, 1) 6= 0. Then assuming the Riemann hypothesisfor L(A, s) (i.e., that L(A, s) 6= 0 for Re(s) > 1), Goldfeld [Gol82] proved that theEuler product for L(A, s), formally evaluated at 1, converges but does not convergeto L(A, 1). Instead, it converges (very slowly) to L(A, 1)/

√2. For further details

and insight into this strange behavior, see [Con03].

Remark 13.1.8. The Clay Math Institute has offered a one million dollar prize fora proof of Conjecture 13.1.5 for elliptic curves over Q. See [Wil00].

Theorem 13.1.9 (Kolyvagin-Logachev). Suppose f ∈ S2(Γ0(N)) is a newformsuch that ords=1 L(f, s) ≤ 1. Then Conjecture 13.1.5 is true for Af .


Theorem 13.1.10 (Kato). Suppose f ∈ S2(Γ1(N)) and L(f, 1) 6= 0. Then Con-jecture 13.1.5 is true for Af .

13.2 Refined Rank Zero Conjecture 125

13.2 Refined Rank Zero Conjecture

Let f ∈ S2(Γ1(N)) be a newform of level N , and let Af ⊂ J1(N) be the corre-sponding abelian variety.

The following conjecture refines Conjecture 13.1.5 in the case L(A, 1) 6= 0. Werecall some of the notation below, where we give a formula for L(A, 1)/ΩA, whichcan be computed up to an vinteger, which we call the Manin index. Note that thedefinitions, results, and proofs in this section are all true exactly as stated withX1(N) replaced by X0(N), which is relevant if one wants to do computations.

Conjecture 13.2.1 (Birch and Swinnerton-Dyer). Suppose L(A, 1) 6= 0.Then

L(A, 1)

ΩA=

#X(A) ·∏p|N cp


By Theorem 13.1.10, the group X(A) is finite, so the right hand side makessense. The right hand side is a rational number, so if Conjecture 13.2.1 is true,then the quotient L(A, 1)/ΩA should also be a rational number. In fact, this istrue, as we will prove below (see Theorem 13.2.11). Below we will discuss aspectsof the proof of rationality in the case that A is an elliptic curve, and at the end ofthis section we give a proof of the general case.

In to more easily understanding L(A, 1)/ΩA, it will be easiest to work withA = A∨

f , where A∨f is the dual of Af . We view A naturally as a quotient of

J1(N) as follows. Dualizing the map Af → J1(N) we obtain a surjective mapJ1(N) → A∨

f . Passing to the dual doesn’t affect whether or not L(A, 1)/ΩA isrational, since changing A by an isogeny does not change L(A, 1), and only changesΩA by multiplication by a nonzero rational number.

13.2.1 The Number of Real Components

Definition 13.2.2 (Real Components). Let c∞ be the number of connectedcomponents of A(R).

If A is an elliptic curve, then c∞ = 1 or 2, depending on whether the graph ofthe affine part of A(R) in the plane R2 is connected. For example, Figure 13.2.1shows the real points of the elliptic curve defined by y2 = x3−x in the three affinepatches that cover P2. The completed curve has two real components.

In general, there is a simple formula for c∞ in terms of the action of complexconjugation on H1(A(R),Z), which can be computed using modular symbols. Theformula is

log2(c∞) = dimF2A(R)[2]− dim(A).

13.2.2 The Manin Index

The map J1(N) → A induces a map J → A on Neron models. Pullback ofdifferentials defines a map

H0(A,Ω1A/Z)→ H0(J ,Ω1

J /Z). (13.2.1)

One can show that there is a q-expansion map

H0(J ,Ω1J /Z)→ Z[[q]] (13.2.2)


-2 -1 0 1 2 3-5

-4

-3

-2

-1

0

1

2

3

4

5

x

y

-2 -1 0 1 2-5

-4

-3

-2

-1

0

1

2

3

4

5

x

z

-3 -2 -1 0 1-5

-4

-3

-2

-1

0

1

2

3

4

5

z

y

FIGURE 13.2.1. Graphs of real solutions to y2z = x3− xz2 on three affine patches

which agrees with the usual q-expansion map after tensoring with C. (For usX1(N)is the curve that parameterizes pairs (E,µN → E), so that there is a q-expansionmap with values in Z[[q]].)

Let ϕA be the composition of (13.2.1) with (13.2.2).

Definition 13.2.3 (Manin Index). The Manin index cA of A is the index ofϕA(H0(A,Ω1

A/Z)) in its saturation. I.e., it is the order of the quotient group

(Z[[q]]

ϕA(H0(A,Ω1A/Z))

)

tor

.

Open Problem 13.2.4. Find an algorithm to compute cA.

Manin conjectured that cA = 1 when dimA = 1, and I think cA = 1 in general.

Conjecture 13.2.5 (Agashe, Stein). cA = 1.

This conjecture is false if A is not required to be attached to a newform, evenif Af ⊂ J1(N)new. For example, Adam Joyce, a student of Kevin Buzzard, foundan A ⊂ J1(431) (and also A′ ⊂ J0(431)) whose Manin constant is 2. Here A isisogenous over Q to a product of two elliptic curves. Also, the Manin index forJ0(33) (viewed as a quotient of J0(33)) is divisible by 3, because there is a cuspform in S2(Γ0(33)) that has integer Fourier expansion at ∞, but not at one of theother cusps.

Theorem 13.2.6. If f ∈ S2(Γ0(N)) then the Manin index c of A∨f can only

divisible by 2 or primes whose square divides N . Moreover, if 4 - N , then ord2(c) ≤dim(Af ).

The proof involves applying nontrivial theorems of Raynaud about exactnessof sequences of differentials, then using a trick with the Atkin-Lehner involu-tion, which was introduced by Mazur in [Maz78], and finally one applies the “q-expansion principle” in characteristic p to deduce the result (see [ASb]). Also,


Edixhoven claims he can prove that if Af is an elliptic curve then cA is only divis-ible by 2, 3, 5, or 7. His argument use his semistable models for X0(p

2), but myunderstanding is that the details are not all written up.

13.2.3 The Real Volume ΩA

Definition 13.2.7 (Real Volume). The real volume ΩA of A(R) is the vol-ume of A(R) with respect to a measure obtained by wedging together a basis forH0(A,Ω1).

If A is an elliptic curve with minimal Weierstrass equation

y2 + a1xy + a3y = x3 + a2x2 + a4x+ a6,

then one can show that

ω =dx

2y + a1x+ a3(13.2.3)

is a basis for H0(A,Ω1). Thus

ΩA =

∫

A(R)

dx

2y + a1x+ a3.

There is a fast algorithm for computing ΩA, for A an elliptic curve, which relies onthe quickly-convergent Gauss arithmetic-geometric mean (see [Cre97, §3.7]). Forexample, if A is the curve defined by y2 = x3 − x (this is a minimal model), then

ΩA ∼ 2× 2.622057554292119810464839589.

For a general abelian variety A, it is an open problem to compute ΩA. However, wecan compute ΩA/cA, where cA is the Manin index of A, by explicitly computing Aas a complex torus using the period mapping Φ, which we define in the next section.

13.2.4 The Period Mapping

LetΦ : H1(X1(N),Z)→ HomC(Cf1 + · · ·+ Cfd,C)

be the period mapping on integral homology induced by integrating homologyclasses on X0(N) against the C-vector space spanned by the Gal(Q/Q)-conjugatesfi of f . Extend Φ to H1(X1(N),Q) by Q-linearity. We normalize Φ so thatΦ(0,∞)(f) = L(f, 1). More explicitly, for α, β ∈ P1(Q), we have

Φ(α, β)(f) = −2πi

∫ β

α

f(z)dz.

The motivation for this normalization is that

L(f, 1) = −2πi

∫ i∞

0

f(z)dz, (13.2.4)

which we see immediately from the Mellin transform definition of L(f, s):

L(f, s) = (2π)sΓ(s)−1

∫ i∞

0

(−iz)sf(z)dz

z.


13.2.5 Manin-Drinfeld Theorem

Recall the Manin-Drinfeld theorem, which we proved long ago, asserts that 0,∞ ∈H1(X0(N),Q). We proved this by explicitly computing (p + 1 − Tp)(0,∞), forp - N , noting that the result is in H1(X0(N),Z), and inverting p + 1 − Tp. Thusthere is an integer n such that n0,∞ ∈ H1(X0(N),Z).

Suppose that A = A∨f is an elliptic curve quotient of J0(N). Rewriting (13.2.4)

in terms of Φ, we have Φ(0,∞) = L(f, 1). Let ω be a minimal differential on A,as in (13.2.3), so ω = −cA · 2πif(z)dz, where cA is the Manin index of A, and theequality is after pulling ω back to H0(X0(N),Ω) ∼= S2(Γ0(N)). Note that whenwe defined cA, there was no factor of 2πi, since we compared ω with f(q) dqq , and

q = e2πiz, so dq/q = 2πidz.

13.2.6 The Period Lattice

The period lattice of A with respect to a nonzero differential g on A is

Lg =

∫

γ

g : γ ∈ H1(A,Z)

,

and we have A(C) ∼= C/Lg. This is the Abel-Jacobi theorem, and the signifi-cance of g is that we are choosing a basis for the one-dimensional C-vector spaceHom(H0(A,Ω),C), in order to embed the image of H1(A,Z) in C.

The integral∫A(R)

g is “visible” in terms of the complex torus representation

of A(C) = C/Lg. More precisely, if Lg is not rectangular, then A(R) may beidentified with the part of the real line in a fundamental domain for Lg, and∫A(R)

g is the length of this segment of the real line. If Lg is rectangular, then

it is that line along with another line above it that is midway to the top of thefundamental domain.

The real volume, which appears in Conjecture 13.2.1, is

ΩA =

∫

A(R)

ω = −cA · 2πi∫

A(R)

f.

Thus ΩA is the least positive real number in Lω = −cA · 2πiLf , when the periodlattice is not rectangular, and twice the least positive real number when it is.

13.2.7 The Special Value L(A, 1)

Proposition 13.2.8. We have L(f, 1) ∈ R.

Proof. With the right setup, this would follow immediately from the fact thatz 7→ −z fixes the homology class 0,∞. However, we don’t have such a setup, sowe give a direct proof.

Just as in the proof of the functional equation for Λ(f, s), use that f is aneigenvector for the Atkin-Lehner operator WN and (13.2.4) to write L(f, 1) as the


sum of two integrals from i/√N to i∞. Then use the calculation

2πi

∫ i∞

i/√N

∞∑

n=1

ane2πinzdz = −2πi∞∑

n=1

an

∫ i∞

i/√N

e2πinzdz

= −2πi

∞∑

n=1

an1

2πine−2πn/

√N

= 2πi

∞∑

n=1

an1

2πine2πn/

√N

to see that L(f, 1) = L(f, 1).

Remark 13.2.9. The BSD conjecture implies that L(f, 1) ≥ 0, but this is unknown(it follows from GRH for L(f, s)).

13.2.8 Rationality of L(A, 1)/ΩA

Proposition 13.2.10. Suppose A = Af is an elliptic curve. Then L(A, 1)/ΩA ∈Q. More precisely, if n is the smallest multiple of 0,∞ that lies in H1(X0(N),Z)and cA is the Manin constant of A, then 2n · cA · L(A, 1)/ΩA ∈ Z.

Proof. By the Manin-Drinfeld theorem n0,∞ ∈ H1(X0(N),Z), so

n · L(f, 1) = −n · 2πi ·∫ i∞

0

f(z)dz ∈ −2πi · Lf =1

cALω.

Combining this with Proposition 13.2.8, we see that

n · cA · L(f, 1) ∈ L+ω ,

where L+ω is the submodule fixed by complex conjugation (i.e., L+

ω = L∩R). Whenthe period lattice is not rectangular, ΩA generates L+

ω , and when it is rectangular,12ΩA generates. Thus n · cA · L(f, 1) is an integer multiple of 1

2ΩA, which provesthe proposition.

Proposition 13.2.10 can be more precise and generalized to abelian varietiesA = A∨

f attached to newforms. One can also replace n by the order of the imageof (0)− (∞) in A(Q).

Theorem 13.2.11 (Agashe, Stein). Suppose f ∈ S2(Γ1(N)) is a newform andlet A = A∨

f be the abelian variety attached to f . Then we have the following equalityof rational numbers:

|L(A, 1)|ΩA

=1

c∞ · cA· [Φ(H1(X1(N),Z))+ : Φ(T0,∞)].

Note that L(A, 1) ∈ R, so |L(A, 1)| = ±L(A, 1), and one expects, of course, thatL(A, 1) ≥ 0.


For V and W lattices in an R-vector space M , the lattice index [V : W ] is bydefinition the absolute value of the determinant of a change of basis taking a basisfor V to a basis for W , or 0 if W has rank smaller than the dimension of M .

Proof. Let ΩA be the measure of A(R) with respect to a basis for S2(Γ1(N),Z)[If ],

where If is the annihilator in T of f . Note that ΩA · cA = ΩA, where cA is theManin index. Unwinding the definitions, we find that

ΩA = c∞ · [Hom(S2(Γ1(N),Z)[If ],Z) : Φ(H1(X0(N),Z))+].

For any ring R the pairing

TR × S2(Γ1(N), R)→ R

given by 〈Tn, f〉 = a1(Tnf) is perfect, so (T/If )⊗R ∼= Hom(S2(Γ1(N), R)[If ], R).Using this pairing, we may view Φ as a map

Φ : H1(X1(N),Q)→ (T/If )⊗C,

so that

ΩA = c∞ · [T/If : Φ(H1(X0(N),Z))+].

Note that (T/If )⊗C is isomorphic as a ring to a product of copies of C, withone copy corresponding to each Galois conjugate fi of f . Let πi ∈ (T/If )⊗C bethe projector onto the subspace of (T/If )⊗C corresponding to fi. Then

Φ(0,∞) · πi = L(fi, 1) · πi.

Since the πi form a basis for the complex vector space (T/If ) ⊗ C, if we viewΦ(0,∞) as the operator “left-multiplication by Φ(0,∞)”, then

det(Φ(0,∞)) =∏

i

L(fi, 1) = L(A, 1),

Letting H = H1(X0(N),Z), we have

[Φ(H)+ : Φ(T0,∞)] = [Φ(H)+ : (T/If ) · Φ(0,∞)]= [Φ(H)+ : T/If ] · [T/If : T/If · Φ(0,∞)]=c∞

ΩA· |det(Φ(0,∞))|

=c∞cAΩA

· |L(A, 1)|,

which proves the theorem.

Remark 13.2.12. Theorem 13.2.11 is false, in general, when A is a quotient ofJ1(N) not attached to a single Gal(Q/Q)-orbit of newforms. It could be modifiedto handle this more general case, but the generalization seems not to has beenwritten down.

13.3 General Refined Conjecture 131

13.3 General Refined Conjecture

Conjecture 13.3.1 (Birch and Swinnerton-Dyer). Let r = ords=1 L(A, s).Then r is the rank of A(Q), the group X(A) is finite, and

L(r)(A, 1)

r!=

#X(A) · ΩA · RegA ·∏p|N cp



13.4 The Conjecture for Non-Modular Abelian Varieties

Conjecture 13.3.1 can be extended to general abelian varieties over global fields.Here we discuss only the case of a general abelian variety A over Q. We follow thediscussion in [Lan91, 95-94] (Lang, Number Theory III), which describes Gross’sformulation of the conjecture for abelian varieties over number fields, and to whichwe refer the reader for more details.

For each prime number `, the `-adic Tate module associated to A is

Ta`(A) = lim←−n

A(Q)[`n].

Since A(Q)[`n] ∼= (Z/`nZ)2 dim(A), we see that Ta`(A) is free of rank 2 dim(A) as aZ`-module. Also, since the group structure on A is defined over Q, Ta`(A) comesequipped with an action of Gal(Q/Q):

ρA,` : Gal(Q/Q)→ Aut(Ta`(A)) ≈ GL2d(Z`).

Suppose p is a prime and let ` 6= p be another prime. Fix any embeddingQ → Qp, and notice that restriction defines a homorphism r : Gal(Qp/Qp) →Gal(Q/Q). Let Gp ⊂ Gal(Q/Q) be the image of r. The inertia group Ip ⊂ Gp isthe kernel of the natural surjective reduction map, and we have an exact sequence

0→ Ip → Gal(Qp/Qp)→ Gal(Fp/Fp)→ 0.

The Galois group Gal(Fp/Fp) is isomorphic to Z with canonical generator x 7→ xp.Lifting this generator, we obtain an element Frobp ∈ Gal(Qp/Qp), which is well-

defined up to an element of Ip. Viewed as an element of Gp ⊂ Gal(Q/Q), theelement Frobp is well-defined up Ip and our choice of embedding Q → Qp. One

can show that this implies that Frobp ∈ Gal(Q/Q) is well-defined up to Ip andconjugation by an element of Gal(Q/Q).

For a Gp-module M , let

M Ip = x ∈M : σ(x) = x all σ ∈ Ip.Because Ip acts trivially on M Ip , the action of the element Frobp ∈ Gal(Q/Q)on M Ip is well-defined up to conjugation (Ip acts trivially, so the “up to Ip”obstruction vanishes). Thus the characteristic polynomial of Frobp on M Ip is well-defined, which is why Lp(A, s) is well-defined. The local L-factor of L(A, s) at pis

Lp(A, s) =1

det(I − p−s Frob−1

p |HomZ`(Ta`(A),Z`)Ip

) .

Definition 13.4.1. L(A, s) =∏

all p

Lp(A, s)

For all but finitely many primes Ta`(A)Ip = Ta`(A). For example, if A = Af isattached to a newform f =

∑anq

n of levelN and p - `·N , then Ta`(A)Ip = Ta`(A).In this case, the Eichler-Shimura relation implies that Lp(A, s) equals

∏Lp(fi, s),

where the fi =∑an,iq

n are the Galois conjugates of f and Lp(fi, s) = (1− ap,i ·p−s + p1−2s)−1. The point is that Eichler-Shimura can be used to show that the

characteristic polynomial of Frobp is∏dim(A)i=1 (X2 − ap,iX + p1−2s).

Theorem 13.4.2. L(Af , s) =∏di=1 L(fi, s).

13.5 Visibility of Shafarevich-Tate Groups 133

13.5 Visibility of Shafarevich-Tate Groups

Let K be a number field. Suppose

0→ A→ B → C → 0

is an exact sequence of abelian varieties over K. (Thus each of A, B, and C is acomplete group variety over K, whose group is automatically abelian.) Then thereis a corresponding long exact sequence of cohomology for the group Gal(Q/K):

0→ A(K)→ B(K)→ C(K)δ−→ H1(K,A)→ H1(K,B)→ H1(K,C)→ · · ·

The study of the Mordell-Weil group C(K) = H0(K,C) is popular in arithmeticgeometry. For example, the Birch and Swinnerton-Dyer conjecture (BSD conjec-ture), which is one of the million dollar Clay Math Problems, asserts that thedimension of C(K)⊗Q equals the ordering vanishing of L(C, s) at s = 1.

The group H1(K,A) is also of interest in connection with the BSD conjecture,because it contains the Shafarevich-Tate group

X(A) = X(A/K) = Ker

(H1(K,A)→

⊕

v

H1(Kv, A)

)⊂ H1(K,A),

where the sum is over all places v of K (e.g., when K = Q, the fields Kv are Qp

for all prime numbers p and Q∞ = R).The group A(K) is fundamentally different than H1(K,C). The Mordell-Weil

group A(K) is finitely generated, whereas the first Galois cohomology H1(K,C) isfar from being finitely generated—in fact, every element has finite order and thereare infinitely many elements of any given order.

This talk is about “dimension shifting”, i.e., relating information about H0(K,C)to information about H1(K,A).

13.5.1 Definitions

Elements of H0(K,C) are simply points, i.e., elements of C(K), so they are rela-tively easy to “visualize”. In contrast, elements of H1(K,A) are Galois cohomologyclasses, i.e., equivalence classes of set-theoretic (continuous) maps f : Gal(Q/K)→A(Q) such that f(στ) = f(σ)+σf(τ). Two maps are equivalent if their differenceis a map of the form σ 7→ σ(P )− P for some fixed P ∈ A(Q). From this point ofview H1 is more mysterious than H0.

There is an alternative way to view elements of H1(K,A). The WC group of Ais the group of isomorphism classes of principal homogeneous spaces for A, wherea principal homogeneous space is a variety X and a map A×X → X that satisfiesthe same axioms as those for a simply transitive group action. Thus X is a twist asvariety of A, but X(K) = ∅, unless X ≈ A. Also, the nontrivial elements of X(A)correspond to the classes in WC that have a Kv-rational point for all places v, butno K-rational point.

Mazur introduced the following definition in order to help unify diverse con-structions of principal homogeneous spaces:


Definition 13.5.1 (Visible). The visible subgroup of H1(K,A) in B is

VisB H1(K,A) = Ker(H1(K,A)→ H1(K,B))

= Coker(B(K)→ C(K)).

Remark 13.5.2. Note that VisB H1(K,A) does depend on the embedding of Ainto B. For example, suppose B = B1 × A. Then there could be nonzero visibleelements if A is embedding into the first factor, but there will be no nonzerovisible elements if A is embedded into the second factor. Here we are using thatH1(K,B1 ×A) = H1(K,B1)⊕H1(K,A).

The connection with the WC group of A is as follows. Suppose

0→ Af−→ B

g−→ C → 0

is an exact sequence of abelian varieties and that c ∈ H1(K,A) is visible in B.Thus there exists x ∈ C(K) such that δ(x) = c, where δ : C(K) → H1(K,A) isthe connecting homomorphism. Then X = π−1(x) ⊂ B is a translate of A in B, sothe group law on B gives X the structure of principal homogeneous space for A,and one can show that the class of X in the WC group of A corresponds to c.

Lemma 13.5.3. The group VisB H1(K,A) is finite.

Proof. Since VisB H1(K,A) is a homomorphic image of the finitely generated groupC(K), it is also finitely generated. On the other hand, it is a subgroup of H1(K,A),so it is a torsion group. The lemma follows since a finitely generated torsion abeliangroup is finite.

13.5.2 Every Element of H1(K,A) is Visible Somewhere

Proposition 13.5.4. Let c ∈ H1(K,A). Then there exists an abelian variety B =Bc and an embedding A → B such that c is visible in B.

Proof. By definition of Galois cohomology, there is a finite extension L of K suchthat resL(c) = 0. Thus c maps to 0 in H1(L,AL). By a slight generalization of theShapiro Lemma from group cohomology (which can be proved by dimension shift-ing; see, e.g., Atiyah-Wall in Cassels-Frohlich), there is a canonical isomorphism

H1(L,AL) ∼= H1(K,ResL/K(AL)) = H1(K,B),

where B = ResL/K(AL) is the Weil restriction of scalars of AL back down to K.The restriction of scalars B is an abelian variety of dimension [L : K] · dimA thatis characterized by the existence of functorial isomorphisms

MorK(S,B) ∼= MorL(SL, AL),

for any K-scheme S, i.e., B(S) = AL(SL). In particular, setting S = A we findthat the identity map AL → AL corresponds to an injection A → B. Moreover,c 7→ resL(c) = 0 ∈ H1(K,B).

Remark 13.5.5. The abelian variety B in Proposition 13.5.4 is a twist of a powerof A.


13.5.3 Visibility in the Context of Modularity

Usually we focus on visibility of elements in X(A). There are a number of otherresults about visibility in various special cases, and large tables of examples in thecontext of elliptic curves and modular abelian varieties. There are also interestingmodularity questions and conjectures in this context.

Motivated by the desire to understand the Birch and Swinnerton-Dyer conjecturemore explicitly, I developed (with significant input from Agashe, Cremona, Mazur,and Merel) computational techniques for unconditionally constructing Shafarevich-Tate groups of modular abelian varieties A ⊂ J0(N) (or J1(N)). For example, ifA ⊂ J0(389) is the 20-dimensional simple factor, then

Z/5Z× Z/5Z ⊂X(A),

as predicted by the Birch and Swinnerton-Dyer conjecture. See [CM00] for exam-ples when dimA = 1. We will spend the rest of this section discussing the examplesof [ASc, AS02] in more detail.

Tables 13.5.1–13.5.4 illustrate the main computational results of [ASc]. Thesetables were made by gathering data about certain arithmetic invariants of the19608 abelian varieties Af of level ≤ 2333. Of these, exactly 10360 have satisfyL(Af , 1) 6= 0, and for these with L(Af , 1) 6= 0, we compute a divisor and multipleof the conjectural order of X(Af ). We find that there are at least 168 such thatthe Birch and Swinnerton-Dyer Conjecture implies that X(Af ) is divisible by anodd prime, and we prove for 37 of these that the odd part of the conjectural orderof X(Af ) really divides #X(Af ) by constructing nontrivial elements of X(Af )using visibility.

The meaning of the tables is as follows. The first column lists a level N and anisogeny class, which uniquely specifies an abelian variety A = Af ⊂ J0(N). Thenth isogeny class is given by the nth letter of the alphabet. We will not discuss theordering further, except to note that usually, the dimension of A, which is given inthe second column, is enough to determine A. When L(A, 1) 6= 0, Conjecture 13.2.1predicts that

#X(A)?=L(A, 1)

ΩA· #A(Q)tor ·#A∨(Q)tor∏

p|N cp.

We view the quotient L(A, 1)/ΩA, which is a rational number, as a single quan-tity. We can compute multiples and divisors of every quantity appearing in theright hand side of this equation, and this yields columns three and four, which area divisor S` and a multiple Su of the conjectural order of X(A) (when Su = S`, weput an equals sign in the Su column). Column five, which is labeled odd deg(ϕA),contains the odd part of the degree of the polarization

ϕA : (A → J0(N) ∼= J0(N)∨ → A∨). (13.5.1)

The second set of columns, columns six and seven, contain an abelian varietyB = Bg ⊂ J0(N) such that #(A ∩ B) is divisible by an odd prime divisor of S`and L(B, 1) = 0. When dim(B) = 1, we have verified that B is an elliptic curveof rank 2. The eighth column A∩B contains the group structure of A∩B, wheree.g., [223022] is shorthand notation for (Z/2Z)2 ⊕ (Z/302Z)2. The final column,labeled Vis, contains a divisor of the order of VisA+B(X(A)).

The following proposition explains the significance of the odd deg(ϕA) column.


Proposition 13.5.6. If p - deg(ϕA), then p - VisJ0(N)(H1(Q, A)).

Proof. There exists a complementary morphism ϕA, such that ϕAϕA = ϕAϕA =[n], where n is the degree of ϕA. If c ∈ H1(Q, A) maps to 0 in H1(Q, J0(N)), thenit also maps to 0 under the following composition

H1(Q, A)→ H1(Q, J0(N))→ H1(Q, A∨)ϕA−−→ H1(Q, A).

Since this composition is [n], it follows that c ∈ H1(Q, A)[n], which proves theproposition.

Remark 13.5.7. Since the degree of ϕA does not change if we extend scalars toa number field K, the subgroup of H1(K,A) visible in J0(N)K , still has orderdivisible only by primes that divide deg(ϕA).

The following theorem explains the significance of the B column, and how it wasused to deduce the Vis column.

Theorem 13.5.8. Suppose A and B are abelian subvarieties of an abelian vari-ety C over Q and that A(Q) ∩ B(Q) is finite. Assume also that A(Q) is finite.Let N be an integer divisible by the residue characteristics of primes of bad re-duction for C (e.g., N could be the conductor of C). Suppose p is a prime suchthat

p - 2 ·N ·#((A+B)/B)(Q)tor ·#B(Q)tor ·∏

`

cA,` · cB,`,

where cA,` = #ΦA,`(F`) is the Tamagawa number of A at ` (and similarly for B).Suppose furthermore that B(Q)[p] ⊂ A(Q) as subgroups of C(Q). Then there is anatural injection

B(Q)/pB(Q) → VisC(X(A)).

A complete proof of a generalization of this theorem can be found in [AS02].

Sketch of Proof. Without loss of generality, we may assume C = A + B. Ourhypotheses yield a diagram

0 // B[p] //

²²

Bp //

²²

B //

²²

0

0 // A // C // B′ // 0,

where B′ = C/A. Taking Gal(Q/Q)-cohomology, we obtain the following diagram:

0 // B(Q)p //

²²

B(Q) //

²²

B(Q)/pB(Q) //

²²

0

0 // C(Q)/A(Q) // B′(Q) // VisC(H1(Q, A)) // 0.

The snake lemma and our hypothesis that p - #(C/B)(Q)tor imply that the right-most vertical map is an injection

i : B(Q)/pB(Q) → VisC(H1(Q, A)), (13.5.2)


since C(A)/(A(Q) +B(Q)) is a sub-quotient of (C ′/B)(Q).We show that the image of (13.5.2) lies in X(A) using a local analysis at each

prime, which we now sketch. At the archimedian prime, no work is needed sincep 6= 2. At non-archimedian primes `, one uses facts about Neron models (when ` =p) and our hypothesis that p does not divide the Tamagawa numbers of B (when` 6= p) to show that if x ∈ B(Q)/pB(Q), then the corresponding cohomology classres`(i(x)) ∈ H1(Q`, A) splits over the maximal unramified extension. However,

H1(Qur` /Q`, A) ∼= H1(F`/F`,ΦA,`(F`)),

and the right hand cohomology group has order cA,`, which is coprime to p.Thus res`(i(x)) = 0, which completes the sketch of the proof.

13.5.4 Future Directions

The data in Tables 13.5.1-13.5.4 could be investigated further.It should be possible to replace the hypothesis that B[p] ⊂ A, with the weaker

hypothesis that B[m] ⊂ A, where m is a maximal ideal of the Hecke algebra T. Forexample, this improvement would help one to show that 52 divides the order of theShafarevich-Tate group of 1041E. Note that for this example, we only know thatL(B, 1) = 0, not that B(Q) has positive rank (as predicted by Conjecture 13.1.5),which is another obstruction.

One can consider visibility at a higher level. For example, there are elementsof order 3 in the Shafarevich-Tate group of 551H that are not visible in J0(551),but these elements are visible in J0(2 · 551), according to the computations in[Ste03] (Studying the Birch and Swinnerton-Dyer Conjecture for Modular AbelianVarieties Using MAGMA).

Conjecture 13.5.9 (Stein). Suppose c ∈ X(Af ), where Af ⊂ J0(N). Thenthere exists M such that c is visible in J0(NM). In other words, every element ofX(Af ) is “modular”.


TABLE 13.5.1. Visibility of Nontrivial Odd Parts of Shafarevich-Tate Groups

A dim Sl Su odd deg(ϕA) B dim A ∩B Vis

389E∗ 20 52 = 5 389A 1 [202] 52

433D∗ 16 72 = 7·111 433A 1 [142] 72

446F∗ 8 112 = 11·359353 446B 1 [112] 112

551H 18 32 = 169 NONE563E∗ 31 132 = 13 563A 1 [262] 132

571D∗ 2 32 = 32 ·127 571B 1 [32] 32

655D∗ 13 34 = 32 ·9799079 655A 1 [362] 34

681B 1 32 = 3·125 681C 1 [32] −707G∗ 15 132 = 13·800077 707A 1 [132] 132

709C∗ 30 112 = 11 709A 1 [222] 112

718F∗ 7 72 = 7·5371523 718B 1 [72] 72

767F 23 32 = 1 NONE

794G 12 112 = 11·34986189 794A 1 [112] −817E 15 72 = 7·79 817A 1 [72] −959D 24 32 = 583673 NONE997H∗ 42 34 = 32 997B 1 [122] 32

997C 1 [242] 32

1001F 3 32 = 32 ·1269 1001C 1 [32] −91A 1 [32] −

1001L 7 72 = 7·2029789 1001C 1 [72] −1041E 4 52 = 52 ·13589 1041B 2 [52] −1041J 13 54 = 53 ·21120929983 1041B 2 [54] −1058D 1 52 = 5·483 1058C 1 [52] −1061D 46 1512 = 151·10919 1061B 2 [223022] −1070M 7 3·52 32 ·52 3·5·1720261 1070A 1 [152] −1077J 15 34 = 32 ·1227767047943 1077A 1 [92] −1091C 62 72 = 1 NONE1094F∗ 13 112 = 112 ·172446773 1094A 1 [112] 112

1102K 4 32 = 32 ·31009 1102A 1 [32] −1126F∗ 11 112 = 11·13990352759 1126A 1 [112] 112

1137C 14 34 = 32 ·64082807 1137A 1 [92] −1141I 22 72 = 7·528921 1141A 1 [142] −1147H 23 52 = 5·729 1147A 1 [102] −1171D∗ 53 112 = 11·81 1171A 1 [442] 112

1246B 1 52 = 5·81 1246C 1 [52] −1247D 32 32 = 32 ·2399 43A 1 [362] −1283C 62 52 = 5·2419 NONE1337E 33 32 = 71 NONE1339G 30 32 = 5776049 NONE1355E 28 3 32 32 ·2224523985405 NONE1363F 25 312 = 31·34889 1363B 2 [22622] −1429B 64 52 = 1 NONE1443G 5 72 = 72 ·18525 1443C 1 [71141] −1446N 7 32 = 3·17459029 1446A 1 [122] −




1466H∗ 23 132 = 13·25631993723 1466B 1 [262] 132

1477C∗ 24 132 = 13·57037637 1477A 1 [132] 132

1481C 71 132 = 70825 NONE1483D∗ 67 32 ·52 = 3·5 1483A 1 [602] 32 ·52

1513F 31 3 34 3·759709 NONE1529D 36 52 = 535641763 NONE1531D 73 3 32 3 1531A 1 [482] −1534J 6 3 32 32 ·635931 1534B 1 [62] −1551G 13 32 = 3·110659885 141A 1 [152] −1559B 90 112 = 1 NONE1567D 69 72 ·412 = 7·41 1567B 3 [4411482] −1570J∗ 6 112 = 11·228651397 1570B 1 [112] 112

1577E 36 3 32 32 ·15 83A 1 [62] −1589D 35 32 = 6005292627343 NONE1591F∗ 35 312 = 31·2401 1591A 1 [312] 312

1594J 17 32 = 3·259338050025131 1594A 1 [122] −1613D∗ 75 52 = 5·19 1613A 1 [202] 52

1615J 13 34 = 32 ·13317421 1615A 1 [91181] −1621C∗ 70 172 = 17 1621A 1 [342] 172

1627C∗ 73 34 = 32 1627A 1 [362] 34

1631C 37 52 = 6354841131 NONE1633D 27 36 ·72 = 35 ·7·31375 1633A 3 [64422] −1634K 12 32 = 3·3311565989 817A 1 [32] −1639G∗ 34 172 = 17·82355 1639B 1 [342] 172

1641J∗ 24 232 = 23·1491344147471 1641B 1 [232] 232

1642D∗ 14 72 = 7·123398360851 1642A 1 [72] 72

1662K 7 112 = 11·16610917393 1662A 1 [112] −1664K 1 52 = 5·7 1664N 1 [52] −1679C 45 112 = 6489 NONE1689E 28 32 = 3·172707180029157365 563A 1 [32] −1693C 72 13012 = 1301 1693A 3 [2426022] −1717H∗ 34 132 = 13·345 1717B 1 [262] 132

1727E 39 32 = 118242943 NONE1739F 43 6592 = 659·151291281 1739C 2 [2213182] −1745K 33 52 = 5·1971380677489 1745D 1 [202] −1751C 45 52 = 5·707 103A 2 [5052] −1781D 44 32 = 61541 NONE1793G∗ 36 232 = 23·8846589 1793B 1 [232] 232

1799D 44 52 = 201449 NONE1811D 98 312 = 1 NONE1829E 44 132 = 3595 NONE1843F 40 32 = 8389 NONE1847B 98 36 = 1 NONE1871C 98 192 = 14699 NONE




1877B 86 72 = 1 NONE1887J 12 52 = 5·10825598693 1887A 1 [202] −1891H 40 74 = 72 ·44082137 1891C 2 [421962] −1907D∗ 90 72 = 7·165 1907A 1 [562] 72

1909D∗ 38 34 = 32 ·9317 1909A 1 [182] 34

1913B∗ 1 32 = 3·103 1913A 1 [32] 32

1913E 84 54 ·612 = 52 ·61·103 1913A 1 [102] −1913C 2 [226102] −

1919D 52 232 = 675 NONE1927E 45 32 34

52667 NONE1933C 83 32 ·7 32 ·72 3·7 1933A 1 [422] 32

1943E 46 132 = 62931125 NONE1945E∗ 34 32 = 3·571255479184807 389A 1 [32] 32

1957E∗ 37 72 ·112 = 7·11·3481 1957A 1 [222] 112

1957B 1 [142] 72

1979C 104 192 = 55 NONE

1991C 49 72 = 1634403663 NONE1994D 26 3 32 32 ·46197281414642501 997B 1 [32] −1997C 93 172 = 1 NONE2001L 11 32 = 32 ·44513447 NONE

2006E 1 32 = 3·805 2006D 1 [32] −2014L 12 32 = 32 ·126381129003 106A 1 [92] −2021E 50 56 = 52 ·729 2021A 1 [1002] 54

2027C∗ 94 292 = 29 2027A 1 [582] 292

2029C 90 52 ·2692 = 5·269 2029A 2 [2226902] −2031H∗ 36 112 = 11·1014875952355 2031C 1 [442] 112

2035K 16 112 = 11·218702421 2035C 1 [111221] −2038F 25 5 52 52 ·92198576587 2038A 1 [202] −

1019B 1 [52] −2039F 99 34 ·52 = 13741381043009 NONE2041C 43 34 = 61889617 NONE2045I 39 34 = 33 ·3123399893 2045C 1 [182] −

409A 13 [93701996792] −2049D 31 32 = 29174705448000469937 NONE2051D 45 72 = 7·674652424406369 2051A 1 [562] −2059E 45 5·72 52 ·72 52 ·7·167359757 2059A 1 [702] −2063C 106 132 = 8479 NONE2071F 48 132 = 36348745 NONE2099B 106 32 = 1 NONE2101F 46 52 = 5·11521429 191A 2 [1552] −2103E 37 32 ·112 = 32 ·11·874412923071571792611 2103B 1 [332] 112

2111B 112 2112 = 1 NONE2113B 91 72 = 1 NONE2117E∗ 45 192 = 19·1078389 2117A 1 [382] 192



A dim Sl Su odd deg(ϕA) B dim A ∩B Vis2119C 48 72 = 89746579 NONE2127D 34 32 = 3·18740561792121901 709A 1 [32] −2129B 102 32 = 1 NONE2130Y 4 72 = 7·83927 2130B 1 [142] −2131B 101 172 = 1 NONE2134J 11 32 = 1710248025389 NONE2146J 10 72 = 7·1672443 2146A 1 [72] −2159E 57 132 = 31154538351 NONE

2159D 56 34 = 233801 NONE2161C 98 232 = 1 NONE2162H 14 3 32 3·6578391763 NONE2171E 54 132 = 271 NONE2173H 44 1992 = 199·3581 2173D 2 [3982] −2173F 43 192 32 ·192 32 ·19·229341 2173A 1 [382] 192

2174F 31 52 = 5·21555702093188316107 NONE2181E 27 72 = 7·7217996450474835 2181A 1 [282] −2193K 17 32 = 3·15096035814223 129A 1 [212] −2199C 36 72 = 72 ·13033437060276603 NONE2213C 101 34 = 19 NONE2215F 46 132 = 13·1182141633 2215A 1 [522] −2224R 11 792 = 79 2224G 2 [792] −2227E 51 112 = 259 NONE2231D 60 472 = 91109 NONE2239B 110 114 = 1 NONE

2251E∗ 99 372 = 37 2251A 1 [742] 372

2253C∗ 27 132 = 13·14987929400988647 2253A 1 [262] 132

2255J 23 72 = 15666366543129 NONE2257H 46 36 ·292 = 33 ·29·175 2257A 1 [92] −

2257D 2 [221742] −2264J 22 732 = 73 2264B 2 [1462] −2265U 14 72 = 72 ·73023816368925 2265B 1 [72] −2271I∗ 43 232 = 23·392918345997771783 2271C 1 [462] 232

2273C 105 72 = 72 NONE2279D 61 132 = 96991 NONE2279C 58 52 = 1777847 NONE2285E 45 1512 = 151·138908751161 2285A 2 [223022] −2287B 109 712 = 1 NONE2291C 52 32 = 427943 NONE2293C 96 4792 = 479 2293A 2 [229582] −2294F 15 32 = 3·6289390462793 1147A 1 [32] −2311B 110 52 = 1 NONE2315I 51 32 = 3·4475437589723 463A 16 [134263127691692] −2333C 101 833412 = 83341 2333A 4 [261666822] −

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